But she goes too far in portraying him as an ally of the National Rifle Association during his time in Congress, where he first served in the House and now serves in the Senate.

Mr. Sanders has certainly cast votes that have upset advocates of gun control. But his overall record on legislation related to guns is not one-sided in the manner that Mrs. Clinton suggested.

In the 1990s, he voted against the Brady Bill, which required background checks on gun purchases. His campaign has said he supported requiring background checks, but voted in opposition because of the five-day waiting period it imposed on handgun purchases.

Asked if she would sign legislation from a Democratic Congress that raised the federal minimum wage to $15, Hillary Clinton did not hesitate.

“Of course I would,” she said. "I have supported the Fight for $15."

Senator Bernie Sanders found that response to be curious.

“I am sure a lot of people are very surprised to learn that you supported raising the minimum wage to 15 bucks an hour,” he said. Mrs. Clinton bristled at his statement, and the candidates proceeded to talk over each other.

Mr. Sanders supports raising the federal minimum wage to $15. But Mrs. Clinton’s position is more complicated.

She has supported raising the federal minimum wage to $12, but she has stopped short of going higher on the federal level, pointing out that economic circumstances vary around the country.

At the same time, she has supported local efforts to raise the minimum wage to $15, which is what Mrs. Clinton was referring to when she said she supported the "Fight for $15."

Mrs. Clinton has long gone after Senator Bernie Sanders of Vermont for his mixed record on gun control. Mr. Sanders has noted the rural nature of his home state, which has some of the least restrictive gun laws in the country.

On Monday, Mrs. Clinton held a campaign event on Long Island on the subject of gun violence, and highlighted the number of guns coming into New York from Vermont that wind up being used in crimes.

That works out to about 8.8 guns for every 100,000 people in Vermont, which is a higher rate than any other state when it comes to guns recovered in New York.

But keep in mind that Vermont has a tiny population – it ranks 49th of the 50 states.

It is worth noting that those numbers are imperfect; as the A.T.F. points out, not all guns used in crimes are traced, and not all guns that are traced are used in crimes.

But looking at the total number of guns that were traced to different states, rather than a per capita figure, offers a much different picture of how guns flow across borders into New York.

By that measure, Vermont ranked 13th among states in 2014, supplying 55 of the 3,187 guns that were recovered in New York and whose origin was traced to another state. In other words, guns traced to Vermont accounted for only 1.7 percent of the out-of-state guns recovered in New York.

The biggest out-of-state source was Virginia, to which 395 guns were traced — seven times the number that came from Vermont, which shares a border with New York. Georgia was next, followed by Pennsylvania, Florida, North Carolina and South Carolina.

The flow of guns into New York from Southern states with weaker gun laws is prevalent enough that there is a term for it. Guns are brought into the state via the “Iron Pipeline,” referring to the Interstate 95 corridor. Here is a handy graphic we published in November that sheds more light on interstate gun trafficking.

Discussing climate change on Monday, Mrs. Clinton cited her “very vigorous record” on the subject. Then she proceeded to express bafflement about a stance she said her opponent had taken.

“I couldn't believe it when Senator Sanders opposed the Paris agreement — the best chance we have to actually reverse climate change and deal with the consequences,” Mrs. Clinton said in an interview on “Capital Tonight,” an upstate New York cable news show.

The Paris agreement, reached in December, commits nearly every nation to take action to combat climate change. Given that Mr. Sanders has made climate change a major issue in his campaign, his supposed opposition would indeed seem odd.

But Mrs. Clinton’s characterization was misleading.

It is true that Mr. Sanders did not warmly embrace the Paris agreement. But his lack of enthusiasm was for the opposite reason that Mrs. Clinton suggested.

“While this is a step forward, it goes nowhere near far enough,” Mr. Sanders said in a statement in December. “The planet is in crisis. We need bold action in the very near future and this does not provide that.”

At a campaign stop in Westchester County, N.Y., on Thursday, Mrs. Clinton faced a question from a Greenpeace activist: In the future, would she reject fossil fuel money in her campaign?

“I have money from people who work for fossil-fuel companies,” Mrs. Clinton responded. Pointing her finger, she added, “I am so sick — I am so sick of the Sanders campaign lying about me. I’m sick of it.”

“The fact of the matter is Secretary Clinton has taken significant sums of money from the fossil fuel industry,” he said on ABC’s “Good Morning America” on Friday. He later said that Mrs. Clinton should apologize for saying his campaign had lied.

We took a look at the numbers.

First of all, Mrs. Clinton’s campaign has not received money from oil, gas or coal companies, because corporations cannot donate to candidates' campaigns.

When it comes to people who work in the oil and gas industry, Mrs. Clinton has indeed raised more money than Mr. Sanders. She has collected about $308,000, compared to about $54,000 for Mr. Sanders, according to the Center for Responsive Politics, which tracks contributions by industry.

But that amount raised by Mrs. Clinton represents less than one-half of 1 percent of her total fund-raising that the Center for Responsive Politics was able to track to specific industries, and the oil and gas industry is not among the top 20 industries that have helped finance her campaign.

Nearly $9 of every $10 donated to presidential candidates in this election cycle from the oil and gas industry has gone to Republicans.

Campaigning in Wisconsin on Friday, Mr. Sanders cited an analysis by Greenpeace that took a broader look at financial support for Mrs. Clinton from the oil, gas and coal industries.

The organization said that a "super PAC" supporting Mrs. Clinton, for instance, had received $3.25 million from donors connected to the fossil fuel industry, though that represents a small piece of what outside groups have raised to support her presidential bid.

Greenpeace also looked at contributions to Mrs. Clinton’s campaign that were donated or bundled by lobbyists who represent the fossil fuel industry, which amounted to about $1.5 million, according to the organization. But that figure still is a small slice of Mrs. Clinton’s fundraising, and it includes lobbyists who represent clients in a range of industries in addition to fossil fuel companies.

Donald J. Trump said Senator Ted Cruz had shifted his views on ethanol to curry votes in Iowa, a major ethanol producer. In 2013, Mr. Cruz, who generally opposes subsidies for industry, supported an immediate end to the government mandate that biofuels like ethanol be mixed into gasoline. In 2014, he said he would support an end to the mandate after several years, but not immediately.

However, since 2014, Mr. Cruz has consistently supported a phaseout of the mandate.

Gov. John Kasich said he would pressure China to use its influence on its neighbor North Korea, resist its aggression in the South China Sea and strengthen United States capabilities to fight Chinese cyberhacking.

Senator Marco Rubio would like to see more legal immigrants to the United States granted green cards on the basis of their skills. He said that most green cards are granted because the immigrant already has family here, acknowledging that members of his own family were allowed to stay in the United States for that reason.

The numbers back him up.

According to data from the Department of Homeland Security, immigrants who obtained green cards on the basis of a family relationship accounted for 66 percent of the total in 2013, for example.

The Social Security Administration projects it has enough money to pay full benefits until 2034. It could remain solvent beyond that date if economic growth strengthens. Relatively modest changes would be sufficient to keep the program solvent: Some combination of higher taxes, smaller benefits or postponing the age of eligibility for future recipients.

After Mr. Putin called Mr. Trump "a very bright and talented man," Mr. Trump effectively returned the compliment. "When people call you brilliant, it's always good, especially when the person heads up Russia," he said on MSNBC's "Morning Joe."

When pressed, Mr. Trump elaborated: "He's running his country, and at least he's a leader, you know, unlike what we have in this country."

Mr. Kasich focused intently on New Hampshire, and he was prolific with his events there. But he inflated the number of town hall-style meetings that he held in the state — in which he takes questions from the audience.

His total count for town halls tops 100, but not by much. He celebrated what his campaign said was his 100th town hall on Feb. 5, just four days before the state's primary.

Mr. Kasich did make other kinds of appearances in New Hampshire that are not included in that tally. A count by New England Cable News put him at 190 total campaign stops.

Mr. Rubio has previously faced criticism during the campaign for missing Senate votes. That’s not unusual for a senator running for president.

But Mr. Trump is correct in saying Mr. Rubio has been the worst offender among current senators.

The Senate has held 370 roll call votes since the beginning of 2015, and Mr. Rubio has missed 148 of them, or 40 percent.

The second-worst offender is Senator Lindsey Graham of South Carolina, a former presidential candidate, who has missed 30 percent of the votes in that period. Senator Ted Cruz of Texas is next on the absenteeism list, having missed 29 percent.

Defending his claim that he would target not just terrorists but also their families, Mr. Trump said that some of the families of the 9/11 hijackers were allowed to leave the United States and return home — presumably to Saudi Arabia — a couple of days before the attacks.

This is not the case, at least according to the 9/11 Commission report.

The commission found that none of the hijackers had any family members in the United States in the days before the attacks.

As he challenged Donald J. Trump's conservative credentials, Ted Cruz highlighted how the businessman had donated to Hillary Clinton 10 times over the years, including writing four checks to her 2008 presidential campaign. The numbers essentially add up.

A look at campaign finance records shows 10 donations from Mr. Trump to Mrs. Clinton's campaigns, with four of them to her presidential bid. But a closer look shows that one of the donations to the presidential campaign was ultimately refunded.

The United States has indeed been dealing with consistent overall trade deficits for decades, and in recent years, the biggest were with China, Japan, Germany and Mexico.

But the United States records trade surpluses with some countries and territories, including Hong Kong, the Netherlands, the United Arab Emirates and Australia, according to the Census Bureau.

(An earlier version of this factcheck inaccurately described the United States's trade balance with Canada. Last year the United States ran a trade deficit with Canada; it is not true that it recorded its first surplus with Canada since 1990.)

Marco Rubio has repeatedly claimed that Donald Trump is not really as self made as he has made it seem, sayng he received an inheritance of at least $100 million. Mr. Trump has called that claim far overstated.

The fact is that it is unclear exactly how much Mr. Trump inherited from his father, Fred Trump Sr., who was in the real estate business. But it is also undeniable that Mr. Trump inherited quite a bit of money.

In January, The New York Times reported that at the release of Fred Trump's will, "the bulk of the inheritance, at least $20 million" -- was to be divided among Donald Trump and his siblings and their descendants.

Fifteen states have held Republican nominating contests thus far, and Mr. Trump has received about 3.4 million votes out of the 9.8 million that have been cast. That means about 66 percent, or two-thirds, have voted for a candidate other than Mr. Trump.

In response to Marco Rubio's claim that Donald J. Trump is the only candidate on the stage who consistently loses to Hillary Clinton in national polls, Mr. Trump retorted that he had beaten Mrs. Clinton plenty.

He hasn't.

Real Clear Politics has assembled a handy list of polling data, dating back to May. Mrs. Clinton has won virtually every one of them. It is true, though, that her margin of victory has diminished since July.

Mr. Kasich, campaigning in Tennessee last weekend, said his rivals for the Republican nomination had overlooked a critical issue at last week’s debate.

“Isn’t it interesting — in the debate the other night, the only person who mentioned jobs was me,” he told voters in Nashville.

In the debate, Mr. Kasich did emphasize the issue of jobs, citing his record in Ohio.

“If we have a president that does that in America, we will get the economic growth, and that is what this country needs,” he said. “Jobs, jobs and jobs, period.”

But a review of the transcript from the debate shows that Mr. Kasich is not giving his opponents quite enough credit, although they were more glancing in their references to jobs and focused less on the subject.

Discussing immigration, Senator Ted Cruz of Texas brought up what he said were the “millions of Americans who are losing their jobs.” He said the Affordable Care Act had also cost millions of jobs, and warned that Donald J. Trump’s ideas about health care would cost even more jobs. And he ended his closing statement by promising to “bring back jobs.”

Senator Marco Rubio of Florida brought up Mr. Trump’s hiring of foreign workers “to take jobs that Americans could have filled,” and he described the Affordable Care Act as “job-killing.”

Mr. Trump said he had hired tens of thousands of people. He talked about countries like China devaluing their currency, and said that American companies cannot compete and American workers lose their jobs.

And, talking about his popularity with Hispanic voters, he said “they know I'm going to bring jobs back from China, from Japan, from so many other places.”

Mr. Rubio, campaigning in Virginia on Sunday, brought up Mr. Trump’s deferments as he argued that Mr. Trump was “not a tough guy.”

Mr. Trump’s draft deferments received some attention earlier in the campaign. He received a medical deferment in 1968, which his campaign described as “minor” and said was “for bone spurs on both heels of his feet.”

Mr. Trump was indeed an athlete as a young man. In a 2000 book about the Trump family, Gwenda Blair wrote about Mr. Trump's playing squash at Fordham University, which he attended before transferring to the University of Pennsylvania. “He never slammed his squash racket into the wall after losing, like some of the other players,” she wrote.

Asked about Mr. Rubio’s claim, his campaign pointed to an article by PolitiFact that mentioned Mr. Trump’s explanation about bone spurs and stated that he was “active in college sports.”

But it is not known whether squash had anything to do with Mr. Trump’s bone spurs. We will revisit Mr. Rubio's claim if evidence emerges that his squash playing took a physical toll.

A spokeswoman for Mr. Trump, Hope Hicks, said only that he had been “very clear about the circumstances of his deferments.”

Mr. Trump was asked why a wall must be built along the Mexican border, but not the Canadian border. “You’re talking about a border that’s many, many, many times longer,” Mr. Trump responded. He went on to say the border was “about four times longer,” and that the Canadian border was “not our big problem.”

But his estimate of the length of the Canadian border is off. The border with Mexico is about 1,900 miles, while the border with Canada is about 5,500 miles. That makes it little under three times as long.

Donald J. Trump, in the midst of an exchange over recent United States foreign policy on Thursday night, said that the world would be "so much better off" if Muammar el-Qaddafi were still in power in Libya — and that he had never advocated for the overthrow of his government.

Not true.

On his video blog in 2011, Mr. Trump advocated for intervention in Libya in no uncertain terms.

We "should go in, we should stop this guy, which would be very easy and very quick," Trump said. "We could do it surgically, stop him from doing it, and save these lives. This is absolutely nuts. We don’t want to get involved, and you’re gonna end up with something like you’ve never seen before.”

As Ted Cruz attacked Donald Trump, he questioned why Republicans would nominate a candidate who agreed with Hillary Clinton on foreign policy and had said that she was "best secretary of state of modern times."

A look at Mr. Trump's past statements shows that the businessman, indeed, has heaped praise on Mrs. Clinton. In a much-cited interview with Fox News in 2012, Mr. Trump called Mrs. Clinton "a terrific woman," continuing: "She really works hard and I think she does a good job. I like her." Mrs. Clinton was secretary of state at the time.

It wasn't immediately clear, though, if Mr. Trump ever went as far as Mr. Cruz stated -- calling Mrs. Clinton the best secretary of state of modern times.

This came up during the last Republican debate, on Feb. 13, in another skirmish between Senator Ted Cruz and Donald J. Trump. Mr. Trump did offer positive words about Planned Parenthood, but they came with a caveat.

“It does do wonderful things, but not as it relates to abortion,” he said of Planned Parenthood. Mr. Trump added that “there are wonderful things having to do with women's health.”

Senator Ted Cruz said, "Democrats bat about a thousand. Just about everyone they put on the court votes exactly as they want. Republicans have batted worse than .500. More than half of the people we put on the court have been a disaster."

There are countless examples of justices appointed by Republican presidents who were or turned out to be liberals: Justices Stevens, Souter, Blackmun, Brennan and Warren. You could make a case for Justices Kennedy and O'Connor. Mr. Cruz would add Justice Roberts. The last example of a Democratic appointee drifting right is Justice Byron White, who was nominated by President John F. Kennedy.

A recent study by the conservative American Enterprise Institute concluded that there was little evidence the Affordable Care Act has led companies to cut back on employment, although it cautioned that the law is new and an impact could be seen in coming years.

Economists do expect that some people will choose to work less now that they can get health insurance without holding a job. The Congressional Budget Office estimates those choices will reduce the work force by the equivalent of about two million full-time workers by 2025.

Senator Ted Cruz injected Donald J. Trump into the amnesty debate, saying that he had "funded" the so-called Gang of Eight bipartisan group of senators that forged compromise immigration legislation in 2013. That group included Senator Marco Rubio. Specifically, Mr. Cruz said that Mr. Trump had donated to two Republicans and three Democrats who were part of that group.

A look at campaign finance records shows that Mr. Trump did indeed make donations to two Republicans — Senators John McCain and Lindsey Graham — and three Democrats — Senators Chuck Schumer, Richard J. Durbin and Robert Menendez.

But Mr. Cruz's claim is overstated because the donations span a period of many years, including during Mr. McCain's 2008 presidential campaign, so it is an overstatement to say that the businessman had funded the Gang of Eight, or that his contributions had a connection to the amnesty debate.

Donald J. Trump's assertion checks out, though Senator Ted Cruz has been endorsed by numerous House members. Among the Republican candidates who remain in the race, almost all of the Senate endorsements have gone to Senator Marco Rubio of Florida. Here’s a handy tracker from FiveThirtyEight that shows how the candidates have fared with endorsements.

Mr. Cruz suggested his lack of endorsements was not such a bad thing. “Donald, if you want to be liked in Washington, that's not a good attribute for a president,” he told Mr. Trump.

Senator Marco Rubio brought up a New York Times article published online Thursday about Donald J. Trump's use of foreign workers, rather than American workers, for certain jobs at Mar-a-Lago, his exclusive Palm Beach club. Mr. Trump responded that local workers did not want the jobs.
As The Times reported, nearly 300 Americans applied or were referred for job openings as waiters, waitresses, housekeepers and cooks at Mar-a-Lago since 2010. Only 17 of the American applicants were hired, roughly 6 percent, according to records Mar-a-Lago submitted to the United States Department of Labor.

In all but a handful of cases, Mar-a-Lago sought to fill the jobs with hundreds of foreign guest workers from Romania and other countries, according to Labor Department records. Trump pursued more than 500 visas for foreign workers at Mar-a-Lago since 2010, the records show.

Mar-a-Lago told the government that the local applicants were not qualified, could not be reached for interviews or did not want the jobs. But several local applicants contacted by The Times said they would have gladly taken the jobs, like Renee L. Seymore, who had worked in a barbecue restaurant, had an interview and “never heard anything back.”

And Tom Veenstra, an official at a local job placement service, told The Times, “We have hundreds of qualified applicants for jobs like those."

Americans spent about $482 billion on Chinese goods last year. The money was not lost. It was spent on things that people presumably wanted, just like money spent in the United States.

It is true that China spends far less on American goods — about $116 billion last year — a trade deficit that raises concerns about the sustainability of the economic relationship between the United States and China.

In an interview on Fox Business Network on Tuesday, Mr. Trump was more forthright in describing the origin of his campaign funds than he has been in some instances in the past. In this particular case, he noted that he had received some contributions.

But Mr. Trump suggested that all those donations were unsolicited, calling them “little contributions sent in by people who love us, which we take — which, frankly, it's more expensive to send them back.” That may be the case for some of his donors, but it is a distortion of how his campaign is being funded.

While Mr. Trump is not on the fund-raising circuit like other candidates and is not relying on interest groups for campaign donations, he wrongly suggested he is not soliciting contributions. Look no further than Mr. Trump’s website. Close to the suggestion “Show Your Support for Donald Trump” is a red button with the word “Donate.”

In one of the testiest exchanges in a debate full of them, Senator Ted Cruz accused Donald J. Trump of not wanting to defund Planned Parenthood, as Mr. Cruz has wanted to do.

Well, Mr. Trump?

It all depends on when he was asked.

In an interview with CNN last summer, Mr. Trump equivocated, saying he would need to take a closer look at the services provided by Planned Parenthood before eliminating funding. He said he was sure the organization does "some things properly and good and that are good for women."

In an interview with Bill O'Reilly on Fox News in September, Mr. Trump sounded a more definitive note: "I would be totally opposed to funding."

Senator Ted Cruz, in attacking Donald J. Trump's conservative credentials, noted that he had contributed to Democrats like John Kerry, Hillary Clinton, Chuck Schumer and Harry Reid in the past. Mr. Cruz's point was that Mr. Trump's past support of Democrats would carry out to his Supreme Court nominations as president.

At least on the contributions, Mr. Cruz was correct. Campaign finance filings show that Mr. Trump has donated to the campaigns of all four.

It is also fair to note, however, that a closer look at Mr. Trump's giving shows that he has supported many Republicans, too.

Senator Ted Cruz is referring to an interview with Senator Marco Rubio on Univision in April.

In that interview, Mr. Rubio said in Spanish that he would not immediately undo President Obama’s program that gives deportation protection to young undocumented immigrants who came to the United States as children. (The program is not an amnesty; it does not provide any lasting immigration status.)

Mr. Rubio did say, however, that the program “will have to end at some point” and “can't be the permanent policy of the United States.”

This is not the first time Mr. Cruz has brought up what Mr. Rubio said on Univision.

We looked into Mr. Rubio’s comments when Mr. Cruz cited them while campaigning in Iowa shortly before the caucuses there. In that case, Mr. Cruz’s claim was not quite right, because he said Mr. Rubio had pledged that he “wouldn’t rescind amnesty."

This time, he spoke more precisely, referring to Mr. Rubio’s first day in office.

Donald J. Trump railed against government waste and fraud, noting that "thousands and thousands" of people are receiving Social Security benefits at age 106 or older, and that they do not exist.

Mr. Trump is essentially correct, based on inspector general audits of the Social Security Administration, even if "thousands and thousands" might be an exaggeration. A 2013 audit said the administration paid 1,546 deceased beneficiaries approximately $31 million. A later audit showed more payments to the deceased. A separate audit, in 2015, found there were about 6.5 million social security number holders age 112 or older for which the agency did not have death information, but it did not say they were all receiving benefits.

But Mr. Kennedy had been nominated in November 1987, before Reagan’s final year. His two previous selections had failed: Robert H. Bork was rejected by the Senate, and Douglas H. Ginsburg withdrew after admitting that he had smoked marijuana.

Senator Ted Cruz says the fight against the Islamic State has been hurt by President Obama's refusal to arm the pesh merga, the Kurdish fighters in Iraq.

Whether or not his diagnosis is correct, he is right on the facts.

According to Aliza Marcus, the author of “Blood and Belief: The P.K.K. and the Kurdish Fight for Independence,” Mr. Obama has been wary of empowering a Kurdistan that could rise up against the Iraqi army or even declare independence — which could upend United States policy in the country.

Ben Carson, who has written a book about the United States Constitution, was asked in the opening minutes of the debate about the duties of the president in nominating a Supreme Court justice. Specifically, what are President Obama's responsibilities in light of Justice Antonin Scalia's death?

The Constitution does not specifically speak to what Mr. Obama should do at this moment. It says that the president has the power, with the "advice and consent of the Senate," to nominate Supreme Court justices. So, Mr. Carson was technically correct that the Constitution does not give specific clarity on matters such as how quickly Mr. Obama should act.

As an aside, Mr. Carson later noted for the fact-checkers out there that he understood that the Constitution gave the president the power to nominate justices, but that was speaking specifically about the current situation.

There’s no question that there has been a big gulf between the views of the Democratic presidential candidates and the Republican ones on the existence of human-caused climate change, with a great deal of skepticism on the Republican side.

Donald J. Trump, for instance, said in 2012 that climate change was invented by the Chinese, and last year, he said simply, “I believe there’s weather.” Senator Ted Cruz of Texas said last year of climate change, “Data and facts don’t support it.”

But Mr. Sanders is wrong to suggest that every Republican candidate shares those views.

“I think the climate is changing,” former Gov. Jeb Bush of Florida said last week in New Hampshire. “It is inconceivable to me that five billion people living on this planet don’t have an impact on that.”

And Gov. John Kasich of Ohio said in November that climate change might be "another one of those reasons why they call me a moderate."

“I do believe there is climate change, and I think that human beings impact it," he said, "but I also don’t know to what degree we impact it."

When Mr. Cruz was asked at Saturday night’s debate whether waterboarding was torture, he said it was not because “under the law, torture is excruciating pain that is equivalent to losing organs and systems.”

He said he would not “bring it back in any sort of widespread use,” but if the United States were facing, say, an imminent terrorist attack, he suggested he would authorize it: “I would use whatever enhanced interrogation methods we could to keep this country safe.”

(That, apparently, was not muscular enough for Donald J. Trump, who has vowed to bring back waterboarding “and a hell of a lot worse” and on Monday used a vulgar term to describe Mr. Cruz’s more restrained stance.)

Mr. Cruz is embracing a view of waterboarding that is rejected by most experts on torture, though it is still defended by some former officials who were involved in the C.I.A. interrogation program under President George W. Bush.

Waterboarding, in which water is poured over the mouth and nose of a bound subject to produce a feeling of suffocation, has generally been considered torture for centuries. It was used during the Inquisition on accused heretics, by Pol Pot's regime in Cambodia on political prisoners, and by the Japanese during World War II on American prisoners. It later became part of the American prosecution of some Japanese perpetrators of war crimes.

United States law does not explicitly prohibit waterboarding, but it appears to be illegal on at least two grounds. Since the end of 2005, the Detainee Treatment Act has prohibited "cruel, inhuman or degrading treatment" of prisoners, presumably including waterboarding. And in November, Congress limited interrogations to techniques listed in the Army Field Manual, which does not allow waterboarding.

United States law defines torture as acts causing "severe physical or mental pain or suffering." In 2002, the Justice Department's Office of Legal Counsel issued a secret opinion that waterboarding as proposed by the C.I.A. would not be torture, because it would not produce pain equivalent to that produced by "serious physical injury, such as organ failure, impairment of bodily function, or even death." That legal opinion was subsequently formally withdrawn by the department, and since then many officials, including President Obama and Leon E. Panetta when he was C.I.A. director, have said they believe waterboarding is torture.

So Mr. Cruz is reviving a definition of torture that is rejected by most human rights, military and intelligence officials.

During an exchange on foreign policy, Hillary Clinton pointed out that Senator Bernie Sanders voted in 1998 to support "a regime change resolution with respect to Iraq."

Mr. Sanders, on the debate stage, indicated that it was a false claim. And when he had a chance to respond, he seemed to blur the facts, making reference to his later opposition to the Iraq war under the Bush administration.

But the facts show that Mrs. Clinton was correct, at least as it relates to 1998. Then, Mr. Sanders voted for a resolution that called for regime change. In fact, the text of the legislation said, "It should be the policy of the United States to support efforts to remove the regime headed by Saddam Hussein from power in Iraq and to promote the emergence of a democratic government to replace that regime."

Debating immigration records, Senator Bernie Sanders said he thought that it was a "good idea" to allow children from Honduras and other violent places in Central America to stay in the United States. But he suggested that Hillary Clinton was less welcoming. "That was not, as I understand it, the secretary's position," he said.

Indeed, Mrs. Clinton in 2014 took a position akin to that of President Obama, telling CNN that children "should be sent back as soon as it can be determined who responsible adults in their families are."

But, on the campaign trail last August, Mrs. Clinton explained that her position was based on sending a message to Central American families that the journey to the United States was too dangerous. "They were robbed, they were raped, they were kidnapped, they were held for ransom by smugglers. So I think it was the responsible message that I — and many others, including the White House — was trying to say to families, 'Do not let your children, your young children, do this.'"

The Bureau of Labor Statistics reports six different unemployment rates, as our colleague Neil Irwin reported this week.

The official unemployment rate, called U-3, stood at 4.9 percent in January. It represents the total number of people who do not have a job but looked for one in the past four weeks.

But the number cited by Senator Bernie Sanders matches up with another unemployment rate, U-6, which is defined more broadly. It counts as unemployed those who do not have a job but looked for one in the past year, as well as people who want to work full time but had to settle for working part time. That rate stood at 9.9 percent in January.

Senator Ted Cruz of Texas has tried to cast one of his rivals, Senator Marco Rubio of Florida, as being too lenient toward undocumented immigrants. Mr. Cruz drew this contrast with Mr. Rubio while campaigning in Iowa shortly before the caucuses there.

In an interview with Jorge Ramos of Univision in April 2015, Mr. Rubio was asked about President Obama’s program to give deportation protection and work permits to young undocumented immigrants who came to the United States as children, known as Deferred Action for Childhood Arrivals, or DACA.

As of Sept. 30, about 700,000 people had been approved for the program, according to federal data. It is not an amnesty, though it is sometimes described it as such; it does not provide any lasting immigration status and can be canceled by the president at any time.

Mr. Rubio said in Spanish that the program “will have to end at some point” but added that he would not “undo it immediately.” (There was some confusion after the interview about the proper translation of his remarks.)

“You can't cancel it from one minute to the next because there are people who are benefiting from it,” he said. “But it will have to end. It can't be the permanent policy of the United States.”

Mr. Rubio said in the interview that he hoped that the program would end as part of a broader immigration overhaul by Congress, including legalization for undocumented immigrants.

In November, he said that the program needed to end even if no such overhaul was achieved, adding that “we should stop new enrollment.”

Recalling that Gov. Chris Christie of New Jersey had quarantined a nurse during the Ebola outbreak, ABC’s Martha Raddatz asked Mr. Christie if he would consider imposing a quarantine on travelers from Brazil because of the Zika virus.

“You bet I would,” he responded, citing the “strong action” he took during the Ebola outbreak.

“She was showing symptoms and coming back from a place that had the Ebola virus active, and she had been treating patients,” he said, explaining that New Jersey had not “picked her up just for the heck of it.”

“We did it because she was showing symptoms,” he said.

But that doesn’t match up with Ms. Hickox’s account.

In an essay describing her quarantine, Ms. Hickox wrote that four hours after landing at Newark Liberty International Airport, a forehead scanner found her with a temperature of 101 degrees. She attributed that reading to her being flushed and upset.

Ms. Hickox wrote that an oral thermometer later found her temperature to be 98.6 degrees, and that her blood was tested for Ebola and came back negative.

The North Koreans launched a ballistic missile test moments before the debate began. Like the North Koreans, several of the candidates had poor aim, starting with Senator Ted Cruz of Texas.

He argued that the North Korean launch was a "direct result" of the failures in the Bill Clinton administration, which he said "led the world in relaxing sanctions against North Korea." Not quite. The 1994 agreement between the Clinton administration and Kim Jong Il — father of the current leader, Kim Jong Un — froze North Korea's plutonium development in exchange for aid. Plutonium development began again after President George W. Bush's administration walked away from the deal. (The North did cheat by beginning a uranium program, but that wasn't covered in the 1994 agreement.)

Most of the North's development of nuclear weapons accelerated in the Bush era. The first North Korean nuclear test was in 2006.

By several measures, Mr. Trump’s claim about the country’s taxes does not check out.

The United States’ total tax revenue as a share of gross domestic product ranked toward the bottom of the pack among the countries that are members in the Organization for Economic Cooperation and Development.

In 2014, according to O.E.C.D. data, the United States’ tax revenue represented 26 percent of gross domestic product. The organization's average was about 34 percent, and the country with the largest share was Denmark, at 51 percent.

The United States was in the middle of the pack for total taxes per capita, according to O.E.C.D. data.

And if you judge by which country has the highest top personal income tax rate, the United States is not close to the highest, either.

Jeb Bush, contrasting private versus public uses of eminent domain, pointed out that once upon a time Donald J. Trump tried to use eminent domain to take a home from an older woman so he could use the property for one of his casinos in Atlantic City. Mr. Trump, during the debate, brushed off the claim.

Mr. Bush is correct: Mr. Trump, with the help of the city, tried to use eminent domain against the woman, Vera Coking, when she refused to sell. He wanted her property to expand Trump Plaza Hotel and Casino. After a three-year saga, Ms. Coking prevailed in State Supreme Court.

Discussing his tough approach to foreign policy and his experience as a federal prosecutor, Gov. Chris Christie of New Jersey claimed that President Obama and former Secretary of State Hillary Clinton "are for paying ransom for hostages."

That's not true. The United States, in contrast with some other countries who have had citizens kidnapped abroad, has a policy of not paying ransom for hostages. In June, Mr. Obama said the government would not shift from this policy. (He did, however, say that the blanket rule “does not prevent communication with hostage takers by our government, the families of hostages or third parties who help these families.”)

As Iowans headed to caucus on Monday night, Senator Ted Cruz’s campaign spread the word that Ben Carson would drop out of the race. Mr. Cruz has since apologized to Mr. Carson over the matter.

Questioned at the debate, Mr. Cruz presented what he said was the source of that information, saying that CNN reported around 6:30 p.m. that Mr. Carson was “quote, taking a break from campaigning” and that its anchors “said it was highly unusual and highly significant.”

He added that CNN did not correct its report until later in the evening.

But CNN did not report that Mr. Carson was dropping out of the race. And the CNN reporter who was credited with breaking the story reported from the outset that Mr. Carson planned to stay in the race.

The timeline goes like this:

At 6:43 p.m. Central Standard Time, CNN’s Dana Bash informed viewers that Mr. Carson “is going to go back to Florida to his home regardless of how he does tonight here in Iowa.”

“He's going to go there for several days,” she added, “and then afterwards, he's not going to go to South Carolina, he’s not going to go to New Hampshire; he's going to come to Washington, D.C., and he's going to do that because the National Prayer Breakfast is on Thursday.”

While the debate was still in progress on Saturday night, CNN disputed Mr. Cruz’s characterization.

“What Senator Cruz said tonight in the debate is categorically false,” Matt Dornic, a spokesman for CNN, said. “CNN never corrected its reporting because CNN never had anything to correct. The Cruz campaign's actions the night of the Iowa caucuses had nothing to do with CNN's reporting. The fact that Senator Cruz continues to knowingly mislead the voters about this is astonishing.”

Senator Marco Rubio of Florida cites the passage of his Hezbollah sanctions act as one of his crucial achievements. But during Saturday's debate, Gov. Chris Christie of New Jersey said that accomplishment was missing an important detail: Mr. Rubio, the governor said, was not present at the Senate to vote for the bill's passage — calling his absence "truancy."

Here are the facts: Mr. Rubio, indeed, proposed the legislation and considers it an achievement. On Nov. 17, 2015, it passed in the Senate by unanimous consent. And while it was not technically a roll call vote, it is true that Mr. Rubio, who has been criticized by his rivals for missing a number of Senate votes last year, was not present the day of the milestone.

It is true that since Gov. Chris Christie took office in 2010, New Jersey’s credit rating has downgraded nine times. But that doesn't mean it has dropped nine levels.

It was downgraded three times each by the three major rating agencies, Standard & Poor’s, Moody's Investors Service and Fitch Ratings. The most recent downgrade was in April 2015, when Moody’s downgraded the state’s general obligation bonds, citing, in part, “the lack of improvement in the state's weak financial position.”

Senator Bernie Sanders is recalling an exchange from 2007, when Hillary Clinton and Barack Obama were running for president. At a debate, Mr. Obama was asked if he would be willing to meet in the first year of his administration, without precondition, with the leaders of Iran, Syria, Venezuela, Cuba and North Korea. He responded that he would.

“I thought that was irresponsible and frankly naïve to say that he would commit to meeting with Chavez and Castro and others within the first year,” she said. “As I said last night, there needs to be a lot of diplomatic effort."

But Mr. Sanders is being imprecise about what exactly Mrs. Clinton said was naïve. She did not say it was naïve to think it was a good idea to talk to America’s enemies; she said it was naïve to commit to meeting with a list of specific leaders.

Emmanuel Saez, an economist at the University of California, Berkeley, found that from 2009 to 2014, 58 percent of the total real family income growth went to the top 1 percent of families. That’s not “almost all” new income. But looking at the period from 2009 to 2012, 91 percent of income gains went to the top 1 percent, according to Mr. Saez.

But Mrs. Clinton cited the vote to suggest that Mr. Sanders’s track record as a liberal was impure. The immigration overhaul was doomed by opposition on both the left and the right, and 15 Democrats in the Senate voted to block the measure.

Mr. Sanders opposed it over the issue of guest workers, arguing that the legislation would drive down wages for American workers. That was a view shared by some in the labor movement, including the A.F.L.-C.I.O., which applauded Mr. Sanders in 2007.

Wall Street increasingly appears to be Public Enemy No. 1 in the Democratic primary contest, and Hillary Clinton said during Thursday's debate that experts had agreed that she had the toughest plan to rein in banks.

A look at the reviews of the plans presented by Mrs. Clinton and Senator Bernie Sanders shows that both campaigns can claim victory.

On Mrs. Clinton's side, her campaign released a statement showing the support for her plan from a number of experts and well-known people, including the economist Paul Krugman, a columnist for The New York Times, and Bill de Blasio, the New York City mayor.

But others have backed Mr. Sanders's proposal as the more strident one. In January, his campaign released a letter signed by 170 "economists and other experts" who felt his plan was superior.

At the Republican presidential debate on Thursday, and again at a campaign stop on Saturday, Mr. Cruz said millions of people had lost jobs and many had lost their doctors and “seen their premiums skyrocket” because of the Affordable Care Act.

To date, researchers have not found significant effects on jobs at the national level. Large economic trends, including the recovery from the recession, may have outweighed any incentives for employers in some industries to limit or reduce the number of workers or their hours, economists say.

A study published this month in the journal Health Affairs, analyzing Census Bureau data, found “little evidence that the Affordable Care Act had caused increases in part-time employment as of 2015.’’

Some people had suggested that the expansion of Medicaid eligibility under the health care law would encourage low-income workers to reduce their hours or stop working because they no longer needed employer-sponsored insurance. A study by the National Bureau of Economic Research found “very little evidence that the Medicaid expansions decreased work effort,” and another study in Health Affairs came to the same conclusion.

When President Obama signed the Affordable Care Act in March 2010, the unemployment rate was 9.9 percent. In December 2015, it was 5 percent.

The health law has been a boon for some people, allowing them to obtain health insurance for the first time. Mr. Cruz is right that some people lost their insurance. Carriers stopped selling policies that did not comply with new federal standards, but many obtained other coverage.

Some people lost their doctors, either because the consumers switched to different insurers or because their doctors were dropped from their health plans. And for some people, premiums have increased substantially, though in many states the rate of increase is slower than it was before the Affordable Care Act.

Campaigning in northeast Iowa on Tuesday, Mrs. Clinton was asked by an attendee if she would consider appointing President Obama to the Supreme Court.

“Wow, what a great idea,” she responded. “Nobody has ever suggested that to me. Wow. I love that. Wow. He may have a few other things to do, but I’ll tell you, that’s a great idea.”

After explaining that, in her view, the Supreme Court has been heading in the wrong direction, she returned to the original suggestion. “I would certainly take that under advisement,” she said. “I mean, he's brilliant, and he can set forth an argument, and he was a law professor. So he’s got all the credentials.”

Mr. Rubio is not a fan of the idea. “That would be a disaster for this country,” he said at the debate.

As she seeks the Democratic nomination, Mrs. Clinton has cast herself as the best person to build on the work of Mr. Obama’s administration.

But Mr. Rubio’s assertion could be taken to suggest that Mrs. Clinton made a campaign pledge to nominate Mr. Obama to the court. That is not the case. Despite her positive reaction, she did not go as far as Mr. Rubio suggested.

New Jersey added about 65,000 private-sector jobs in 2015, which amounted to an increase of about 1.9 percent from the previous year, according to the federal Bureau of Labor Statistics. That was New Jersey’s largest annual increase since 2000, when the growth in jobs was a hair larger.

Gov. Chris Christie was aggressive, though, in taking credit for those job gains, saying of the increase, “That’s because we put conservative policies in place.” Governors often take credit for — or receive blame for — the change in employment in their states, but those job changes are affected by a multitude of factors.

Senator Marco Rubio is probably overly optimistic when he asserts that countries that lined up with the United States on the Iran nuclear deal would be willing to rip it up and sever all economic ties with Iran. Unless there were evidence of wholesale cheating by Iran, most analysts said that if the United States abrogated the deal, it would find itself isolated, even among its European allies.

As Chris Christie fielded a question about racial profiling and the San Bernardino attacks, he said that the neighbors of the shooters knew that they "had weapons, and they knew that they were talking about trying to take our country and attack it," Mr. Christie said. "That's not profiling."

But the moderator Megyn Kelly interupted Mr. Christie to tell him that he had overstated the facts and that neighbors witnessed men coming and going and saw deliveries, but that they did not think that was enough to call the police because they did not want to profile.

So, did Mr. Christie overstate the facts? Yes, according to news media reports, which include interviews with neighbors who mentioned Middle Eastern men and deliveries, but no one who claimed to know of an attack being planned.

Senator Marco Rubio criticized cap-and-trade regulations, which set overall emissions limits and let companies buy and sell the right to pollute. He also pledged that "there will never be any cap-and-trade" in the United States should he be elected president.

In fact, both the federal government and many state governments already use cap-and-trade systems to control some kinds of emissions, including nitrogen oxide and sulfur dioxide.

Mr. Rubio probably meant that he is opposed to using a cap-and-trade system to regulate carbon emissions. But his view that doing so would be economically destructive is at odds with the views of many economists, both conservative and liberal, who regard it as the most economically sensible approach to the problem.

Mr. Rubio is correct that ISIS has been innovative in its financing and in how its fighters penetrate other countries. But he exaggerates in saying it poses an "unprecedented" threat, in part because the United States has become more effective in protecting itself from a domestic terrorist attack on the scale of Al Qaeda's 9/11 attack.

Census and other data show real median household income in 2014 indeed was nearly $4,000 below its peak before the great recession began in December 2007. But that downturn began nearly a year before Barack Obama took office as president. Wage growth has been stagnant for years, but the 2007-9 recession and high unemployment in its wake depressed income further. Yet monthly median household income has been rising since the third quarter of 2011, in Mr. Obama's first term, according to Sentier Research.

In his opening of the debate, Senator Ted Cruz mocked Donald J. Trump's absence with some playful name-calling of himself and his rivals. But he also said that he has not actually personally insulted Mr. Trump as the campaign has become increasingly contentious.

While Mr. Cruz's personal attacks have not risen to the level of Mr. Trump's "birther" claims, the senator from Texas has surely gone on the offensive. For example, in one talk this month, Mr. Cruz poked Mr. Trump for having a lot of "nervous energy" and for reacting badly to negative polls. And then Mr. Cruz discussed Mr. Trump's propensity to post his feelings on Twitter. “I think in terms of a commander in chief, we ought to have someone who isn’t springing out of bed to tweet in a frantic response to the latest polls,” Mr. Cruz said.

Military analysts largely dismiss carpet-bombing as a tool against an enemy like the Islamic State because its fighters are mingled among civilians in cities and towns in northern Syria and Iraq. The Obama administration has carried out hundreds of targeted airstrikes on Islamic State positions. An intensified military campaign would probably require ground troops, something that neither President Obama nor Senator Ted Cruz support.

That’s not what Mr. Sanders is proposing, though the senator previously suggested that such a tax rate would not be out of the question.

Mr. Sanders does want to raise taxes on the wealthy — in fact, that is a cornerstone of his campaign, which is focused in large part on the big gap between the rich and the poor.

The current top marginal income tax rate is 39.6 percent, which for 2016 applies to income earned by single filers over $415,050, and to income over $466,950 by married taxpayers filing jointly. That does not mean all of those taxpayers’ income is taxed at 39.6 percent; rather, only their income over those amounts is taxed at that rate.

Mr. Sanders has proposed several new tax brackets for high earners, topping out at 52 percent on income above $10 million. (He has proposed other tax changes as well, including what he calls an income-based “premium” to help pay for his health care plan.)

But Mr. Trump is not pulling figures out of thin air when he mentions a 90 percent tax rate, nor is he the only Republican candidate to have suggested Mr. Sanders wanted to raise taxes that high.

In an interview with CNBC in May, Mr. Sanders mentioned that during the Eisenhower administration, the highest marginal tax rate was around 90 percent.

“When you think about 90 percent, you don't think that's obviously too high?” he was asked by CNBC’s John Harwood, who is also a contributor to The New York Times.

“No,” Mr. Sanders responded.

But at a debate in November, Mr. Sanders said he would not seek to raise taxes as high as they were under Eisenhower.

Senator Marco Rubio of Florida has been criticized for being frequently absent from the Senate while he runs for president.

At a town hall meeting last week in Plymouth, N.H., a voter told Mr. Rubio that he did not mind that the senator had missed votes in order to pursue his White House bid. But Mr. Rubio still felt compelled to defend himself.

In citing his own attendance record, Mr. Rubio used a time period — his five-year Senate career — that paints him in a favorable light.

In that period, he has missed about 13 percent of votes. That’s not exactly something to brag about: When it comes to the share of votes missed in a senator’s career, the median is 1.7 percent among senators currently in office, according to GovTrack.

Looking at Mr. Rubio’s attendance more recently gives a very different picture. In the second half of 2015, Mr. Rubio missed 67 of 119 votes, which works out to 56 percent.

It doesn’t make much sense to compare his career attendance with the attendance of Barack Obama and Hillary Clinton when they were running for president.

But Mr. Rubio is on target in his point that Mr. Obama and Mrs. Clinton were frequently absent from the Senate during their 2008 presidential bids.

For a more useful comparison, we looked up their attendance for the second half of 2007. Mr. Obama missed 70 percent of votes, while Mrs. Clinton missed 48 percent.

Mr. Trump likes to declare that the United States is getting ripped off by China. This month, he proposed (and then denied proposing) a 45 percent tariff on Chinese goods.

But in an interview on Fox News on Wednesday, he significantly exaggerated the United States trade deficit with China when he put the number at $500 billion a year.

The trade deficit for goods and services has been growing for years, and hit $315 billion in 2014, the last full year for which the Commerce Department has published data. Through the third quarter of 2015, the deficit totaled $255 billion, according to seasonally adjusted data.

Mr. Sanders has made crusading against Wall Street a major part of his campaign.

At Sunday night’s debate, he cited Goldman Sachs as an example of “how corrupt this system is,” and portrayed its chief executive, Lloyd C. Blankfein, as eager to roll back entitlements, despite his own vast wealth.

But Mr. Sanders gave an incomplete picture of Mr. Blankfein’s views.

In an interview with CBS News in November 2012 that the Sanders campaign said was the basis for the senator's remarks, Mr. Blankfein said that “you’re going to have to undoubtedly do something to lower people’s expectations — the entitlements, and what people think that they’re going to get.” He said that in general, entitlements like Social Security needed to be “slowed down and contained,” since “we can’t afford them.”

At the time, President Obama and Congress were trying to avert the so-called fiscal cliff. Mr. Blankfein was part of the Campaign to Fix the Debt, a group backed by prominent business leaders that urged lawmakers to reach a deal on deficit reduction.

But Mr. Sanders left out what else Mr. Blankfein said in that interview.

“In the long run there has to be more revenue, and of course the burden of that revenue will be disproportionately taken up by wealthier people,” Mr. Blankfein said. “That’s just logical.”

He said that necessitated “more taxes on wealthier people” — a view that Mr. Sanders, of all people, has espoused in his presidential campaign.

Senator Bernie Sanders said that the United States must transform its energy system away from fossil fuels and toward energy-efficiency and sustainable energy. "I've got the most comprehensive legislation in the Senate to do that," he said.

Senator Bernie Sanders's "Climate Protection and Justice Act" would create a tax on emissions of planet-warming carbon emissions, starting at $15 per ton of pollution in 2017 and rising to $73 per ton in 2035, along the lines of the recommendations of many leading environmental economists. It would also create a detailed program for rebating those revenues to economically vulnerable populations who would be adversely affected by the subsequent hit to the fossil fuel industries. Some other liberal Democratic senators, like Sheldon Whitehouse of Rhode Island and Brian Schatz of Hawaii, have also introduced bills to tax carbon dioxide, but Mr. Sanders's bill is the broadest and the most detailed.

During an interview with Mr. Sanders on a radio program in July 2011, a caller said he wished Mr. Sanders would consider running for president, and then asked Mr. Sanders how to get the country back on track.

He responded by citing disappointment with Mr. Obama:

“I think there are millions of Americans who are deeply disappointed in the president, who believe that with regard to Social Security and a number of other issues, he said one thing as a candidate, and is doing something very much else as a president, who cannot believe how weak he has been, for whatever reason, in negotiating with Republicans, and there’s deep disappointment.”

“One of the reasons the president has been able to move so far to the right is that there is no primary opposition to him,” Mr. Sanders added. “And I think it would do this country a good deal of service if people started thinking about candidates out there to begin contrasting what is a progressive agenda as opposed to what Obama is doing.”

Similarly, in an interview the next month, Mr. Sanders said he expected Mr. Obama to be the Democratic nominee, but thought it would be good for the Democratic Party if the incumbent president faced pressure over his record.

“If you’re asking me, do I think at the end of the day that Barack Obama is going to be the Democratic candidate for president in 2012? I do,” Mr. Sanders said. “But do I believe that it is a good idea for our democracy, and for the Democratic Party — and I speak, by the way, as an independent — that people start asking the president some hard questions about why he said one thing during his previous campaign and is doing another thing today on Social Security, on Medicare? I think it is important that that discussion take place.”

Health spending in the United States is higher than it's ever been, so Hillary Clinton's statement on its face is inaccurate.

It is possible she misspoke: It is true that health spending growth was at its lowest level in a half-century in 2013. The difference is important. Health spending growth is a measure of how much more the United States spends on health care in one year compared with the previous year. Even as health spending growth slowed down substantially, overall spending was still on the rise. (The 2014 numbers look slightly less rosy than the rates in 2013.) The improving economy and the coverage expansion of the Affordable Care Act have nudged spending growth back up, though it remains low by historical standards.

Whether the Affordable Care Act should get any credit for this change is a highly disputed matter. The recent, historic slowdown in health spending began before the health law passed. The Affordable Care Act may have helped, but it probably shouldn't get the full credit.

No matter how you measure it, the United States pays more for health care than other developed democracies. Mr. Sanders's number was a slight exaggeration, but basically right. According to the Organization for Economic Cooperation and Development, health care in the United States cost about 2.7 times as much per capita as care in Britain in 2012. (That would be almost "three times as high," not "three times more.") Mr. Sanders also said the U.S. spent 50 percent more than the French, which actually understated the difference; the U.S. spends twice as much per capita as the system in France. Despite those high costs, a smaller percentage of Americans have health insurance when compared to residents of those European countries.

France and Britain aren't outliers, either. Health spending per capita in the United States is substantially higher than the costs of every other country in the OECD.

Dylann Roof, the man suspected of killing nine worshipers at a South Carolina church last year, was allowed to buy a .45-caliber handgun despite having previously admitted to drug possession. That should have prevented him from buying a gun, but errors delayed the process. Under federal law, if there has been no denial within three days, the buyer is presumed to be cleared to get a gun. That's what happened in Mr. Roof's case, and Hillary Clinton and gun-control advocates have taken to calling it the "Charleston loophole." (Tonight's debate is in Charleston.)

That provision has been on the books since the passage of the Brady Handgun Violence Prevention Act in 1993. In some versions of the bill, the time period was seven days; in others, it was five. In November 1993, the House approved an amendment reducing it to one day.

A Senate compromise later raised it to three days, where it has stood since, but Mr. Sanders supported giving background checkers even less time to work before guns could automatically be purchased.

Mr. Roof was one of thousands of prohibited buyers who legally bought firearms over the past decade because of the three-day waiting time provision. It is worth noting, however, that the three-day limit was only one factor in Mr. Roof's case. Mistakes by federal and local authorities created the delay in the first place.

When Donald J. Trump visited the editorial board of The New York Times last week, he fielded a question about how he would pressure China if he were president. He suggested imposing a 45 percent tax on Chinese goods brought into this country.

“I would tax China coming in — products coming in. I would do a tariff. And they do it to us. We have to be smart. I’m a free trader. I’m a free trader. And some of the people would say, ‘Oh, it’s terrible.’ I’m a free trader. I love free trade. But it’s got to be reasonably fair. I would do a tax, and the tax — let me tell you what the tax should be. The tax should be 45 percent."

Mr. Trump added that such a tax would be equivalent "to some of the kind of, you know, devaluations" that China has done to its currency.

There is little doubt that manufacturing employment in the United States has declined because we are importing more stuff from China. A 2013 study estimated that the United States lost almost one million factory jobs between 2000 and 2007 because of increased trade with the Asian nation.

The real question is whether those job losses are a bad thing. The authors of the 2013 study argued that the United States benefits from its relationship with China. Consumers save money on products, and workers are able to find jobs in other industries. Other economists, however, see evidence that many workers who lose jobs suffer permanent damage.

The importance of currency devaluation is also a matter of dispute. Some experts see evidence that China has taken advantage of the United States by suppressing the value of its currency, making its exports relatively cheap for American consumers and American imports relatively expensive for Chinese consumers. However, by most accounts, the extent of this manipulation has declined in recent years — and even without any manipulation, China would still enjoy a significant advantage in manufacturing, thanks to its relatively low labor costs.

Senator Marco Rubio's charge that President Obama has played down the threat of the Islamic State ends up inflating the threat from the group. Several organizations have concluded that the Islamic State targeted Yazidis with a campaign of genocide, but there is no evidence that it so targeted Christians.

Correction: An earlier version of this fact check imprecisely described the Islamic State's record of treatment of religious minorities in areas where it has operated. Several organizations have concluded there is evidence the group carried out a campaign of genocide against the Yazidis, as Mr. Rubio asserted; it was not accurate to say that there is no evidence to support that claim. The statement was made in a Republican presidential debate in January, and the matter was recently brought to The New York Times's attention by several readers.

Gov. Chris Christie said President Obama and Hillary Clinton failed to enforce a red line in threatening to use military force if President Bashar al-Assad of Syria used chemical weapons against his own people. And he said Mrs. Clinton called Mr. Assad a reformer.

Gov. Chris Christie accused President Obama and Hillary Clinton of failing to enforce the president's red line on Syria's use of chemical weapons. Mrs. Clinton had left the State Department eight months before Mr. Obama made the decision not to use military force. Mr. Christie also noted that Mrs. Clinton referred to President Bashar al-Assad as a reformer. She said, in early 2011 before the violence escalated in Syria, that Mr. Assad was viewed by some as a reformer.

Senator Ted Cruz, playing up Donald J. Trump's New York roots (and possibly paying him back for saying "not a lot of evangelicals come out of Cuba"), said that not a lot of conservatives come from Manhattan.

While not known as a hotbed of social conservatism, there are clearly more than a few who would consider themselves conservatives in the economic sense, at least. A look at the voter registrations in Manhattan shows that while Democrats greatly outnumber Republicans, there are certainly more than a few of the latter.

As of Nov. 1, there were 613,634 active registered Democrats and 83,970 active registered Republicans in Manhattan, according to the state elections board.

In 1994, Mr. Christie was running to be a Morris County freeholder, the equivalent of a county commissioner. At the time, Democrats challenged Republican freeholders to restore $35,000 in annual funding for Planned Parenthood of Central and Greater Northern New Jersey, according to an article in The Star-Ledger of Newark.

"I support Planned Parenthood privately with my personal contribution, and that should be the goal of any such agency, to find private donations,” Mr. Christie was quoted as saying.

Jeb Bush's pledge that he would move the United States Embassy from Tel Aviv to Jerusalem as a signal of toughness toward Iran would run counter to American policy and practice in presidential administrations, Democratic and Republican, including that of his brother George W. Bush. Jerusalem's status as the capital of Israel remains disputed.

As Gov. Chris Christie faced an attack on his conservative credentials, he denied the contention that he had supported Sonia Sotomayor's Supreme Court nomination. His deflection came during an exchange with Senator Marco Rubio.

But the truth is that Mr. Christie, as a Republican candidate for governor of New Jersey in 2009, sent out a statement backing the choice of Ms. Sotomayor. "After watching and listening to Judge Sotomayor's performance at the confirmation hearings this week, I am confident that she is qualified for the position of associate justice of the U.S. Supreme Court," Mr. Christie wrote.

The first problem with Gov. John Kasich's claim is that middle-class wages have stagnated for the past few decades, while the Federal Reserve has kept interest rates low for the past few years.

Moreover, consider the counterfactual. The Fed pushed interest rates to historically low levels in the aftermath of a devastating recession. Most economists agree that doing so was a necessary measure to revive economic growth. If it had not done so, a lot of American households would likely be making a lot less money.

Senator Ted Cruz is conflating two types of financial reports. One is the annual personal disclosure in which members of Congress list their assets and income. On those reports, he listed the bank loans — without indicating that they were used for his campaign.

But it is the other type of report, which campaigns must file with the Federal Election Commission, that is at issue. On those reports, Mr. Cruz failed to disclose any bank loans he obtained to finance his Senate race.

So, while it is true that Mr. Cruz listed the loans in his Senate reports, he has never disclosed in any report that they were used for his campaign. Federal election laws require candidates to disclose such loans, and failing to do so has led, in some case, to investigations and fines.

Some family members of the victims of the Benghazi attacks have accused Hillary Clinton of offering a misleading account of the attacks in her conversations with them. But other families members have said that she chose her words carefully and did not make statements that were contradicted by subsequent disclosures. The exchanges were private, and their accounts are contradictory, so it is difficult to know exactly what she said to the families.

President Obama highlighted the good news about the economy in his State of the Union address. The unemployment rate has fallen from 10 percent to 5 percent, and job growth has been steady.

But Senator Ted Cruz is right to point out that big problems remain. The share of adults with jobs remains close to the lowest level in decades and, as Mr. Cruz also noted, wages for many workers have stagnated.

In December, 59.5 percent of adults over 16 had jobs. The number has climbed slightly since the recession, so Mr. Cruz is not literally correct, but he has the basic point right.

Indeed, one reason the unemployment rate has fallen so quickly is that it only counts people looking for jobs. Millions of Americans have stopped looking, and that is a big problem for the economy.

Gov. Chris Christie and Senator Ted Cruz both hammered President Obama for allowing Iran to grab and humiliate United States sailors. In fact, the sailors acknowledged that they drifted into Iranian waters.

Mr. Rubio has tried to gain the upper hand with Mr. Cruz on the issue of national security, assailing Mr. Cruz’s vote in favor of legislation that put an end to the National Security Agency’s bulk collection of domestic phone records. That program came to light in 2013 through the revelations by Edward J. Snowden, the former N.S.A. contractor.

On Sunday, on “This Week,”​ on ABC, ​ Mr. Rubio was asked about his criticism of Mr. Cruz over the issue of domestic surveillance. He responded in part by bringing up Mr. Snowden, whom he called a traitor.

“We cannot afford to have a commander in chief who thinks people like Edward Snowden are doing a good public service,” Mr. Rubio said.

After Mr. Snowden’s identity became public in 2013, Mr. Cruz declined to label him as a patriot or a traitor. Mr. Cruz did offer some praise for Mr. Snowden, but with qualifications.

“If it is the case that the federal government is seizing millions of personal records about law-abiding citizens, and if it is the case that there are minimal restrictions on accessing or reviewing those records, then I think Mr. Snowden has done a considerable public service by bringing it to light,” Mr. Cruz said at an event hosted by TheBlaze, according to the website​​.

But that was not all Mr. Cruz said on the subject, as Mr. Rubio’s selective quotation might suggest. He also said this: “If Mr. Snowden has violated the laws of this country, there are consequences to violating laws and that is something he has publicly stated he understands and I think the law needs to be enforced.”

Since those quotes are from more than two years ago, we asked Mr. Cruz’s campaign how he now assessed Mr. Snowden. In a statement, Mr. Cruz took a very different tone, saying​,​ "​It is now clear that Snowden is a traitor, and he should be tried for treason.”

He pointed to his remark in 2013 that Mr. Snowden should be prosecuted if he broke any laws. "Today, we know that Snowden violated federal law, that his actions materially aided terrorists and enemies of the United States, and that he subsequently fled to China and Russia," he said. "Under the Constitution, giving aid to our enemies is treason."

In an interview with Mr. Trump on Tuesday, Fox Business Network’s Neil Cavuto predicted that Mr. Trump’s rivals were “going to have guns ablazing” for him at Thursday night’s debate, citing the focus on his “45 percent tariff idea for China.”

Mr. Trump immediately sought to distance himself from that idea.

“Well, I don't have a 45 — I'm just saying, I’m just saying that we have to take a tough stance on China because I never said — I don't even know where the 45 percent came from,” he said. “But I said that is the equivalent of what they’ve done with respect to their devaluations. But China has to pay a price if they don't start, you know, behaving, because they're killing us on trade.”

The 45 percent figure came from Mr. Trump himself.

In a meeting with members of the editorial board of The New York Times on Jan. 6, Mr. Trump was asked what he would do as president to exert pressure on China. He responded by saying he would impose a tariff on Chinese goods brought into the United States, which The Times reported last week. Here’s what Mr. Trump said:

“I would tax China coming in — products coming in. I would do a tariff. And they do it to us. We have to be smart. I’m a free trader. I’m a free trader. And some of the people would say, ‘Oh, it’s terrible.’ I’m a free trader. I love free trade. But it’s got to be reasonably fair. I would do a tax, and the tax — let me tell you what the tax should be. The tax should be 45 percent."

Mr. Trump went on to say that such a tax would be equivalent "to some of the kind of, you know, devaluations" that China has done to its currency, and that the Chinese "cannot believe that we haven't done this yet."

After President Obama said last week that he would not support any Democrat who does not favor new gun control measures, Mrs. Clinton has tried to call attention to Mr. Sanders’s vote in 2005 for a bill that shields gun manufacturers and dealers from liability lawsuits. Mr. Sanders was a House member at the time; Mrs. Clinton, then a senator from New York, voted against the bill.

On MSNBC’s “Hardball” on Friday, Mrs. Clinton claimed that the legislation said “no one can sue a gun maker or a gun seller,” and on CBS’s “Face the Nation” on Sunday, she said it provided the gun industry with “absolute immunity from any kind of responsibility or liability.”

The bill, the Protection of Lawful Commerce in Arms Act, followed a flurry of lawsuits against the industry by municipalities. It shields gun manufacturers, sellers and their trade associations from liability lawsuits “resulting from the criminal or unlawful misuse” of firearms and ammunition.

When President George W. Bush signed the measure into law, the National Rifle Association called it “the most significant piece of pro-gun legislation in 20 years.”

Though its protections are quite expansive, the law specifies six exceptions to the prohibition on civil actions. For example, a victim can sue a dealer who has been convicted of selling a firearm knowing that it would be used to commit a violent crime.

Suits are also allowed in matters involving injuries or property damage resulting from manufacturing or design defects, as long as the firearm in question was not used in a criminal act. Another exception allows lawsuits when a gun maker or seller “knowingly violated a state or federal statute applicable to the sale or marketing of the product.”

These exceptions are narrow, and they do provide broad protections to the gun industry, as Mrs. Clinton suggests. Whether the exceptions would ultimately prove meaningful was a subject of debate when Congress passed the legislation in 2005.

“The aim of this bill is clear: to allow legitimate lawsuits against a manufacturer when the legal principles to do so are present,” Senator John Thune, Republican of South Dakota, said at the time.

Senator Jack Reed, Democrat of Rhode Island, took the opposite view: “These exceptions in the bill have been carefully crafted to prevent lawsuits, not to enable appropriate lawsuits to go forward.”

Mr. Lytton, the law professor, said the law "pretty much shut down attempts to use the court system in a broad way to try to push for greater industry self-regulation of marketing and distribution of firearms."

But the exceptions have not put a complete halt to litigation.

In 2008, a lawyer for the Brady Center to Prevent Gun Violence wrote that “reports of the demise of gun liability cases were premature,” adding that “well-pleaded, carefully crafted cases can still proceed against irresponsible gun companies.”

How courts interpret the exceptions is still unfolding, and lawyers have tried to craft lawsuits that could meet their criteria.

As one example, families of some of the victims of the 2012 mass shooting at Sandy Hook Elementary School in Newtown, Conn., are suing Remington Outdoor, the maker of the Bushmaster rifle used in the shooting, among other defendants.

The families are arguing that the gun maker knew, or should have known, that selling assault rifles to civilians “posed an unreasonable and egregious risk of physical injury,” and that its conduct violated the Connecticut Unfair Trade Practices Act.

Remington Outdoor is seeking the suit’s dismissal. The 2005 law, it said in court papers last month, “was enacted to protect firearm manufacturers against the very claims plaintiffs make in this case.”

Mr. Rubio, a senator from Florida, and Mr. Christie, the New Jersey governor, are attacking each other with a notable intensity, and in an interview Thursday on Fox Business Network, Mr. Rubio asserted that Mr. Christie “has done a number of things that are very similar to the Obama agenda,” giving several examples. We looked into his claims.

“The fact of the matter is Chris Christie has supported Common Core and in fact has bad-mouthed Republicans that oppose it.”

Mr. Christie was previously a supporter of the Common Core education standards. “This is one of those areas where I’ve agreed more with the president than not,” he said in 2013.

But Mr. Christie had plenty of company: More than 40 states adopted the Common Core standards.

Mr. Christie’s bad-mouthing was also mild, particularly by his standards. He attributed some of the Common Core opposition among congressional Republicans to “that knee-jerk reaction that’s happening in Washington right now that if the president likes something, the Republicans in Congress don’t,” and vice versa.

Mr. Christie has since changed his position on Common Core, which has become an object of scorn among conservatives who view it as an example of government overreach. In May, Mr. Christie said that Common Core was “simply not working.”

In 1993, Mr. Christie, then a 30-year-old lawyer, announced that he would try to unseat a Republican state senator.

"The issue which has energized me to get into this race is the recent attempt by certain Republican legislators to repeal New Jersey's ban on assault weapons," Mr. Christie said in a statement, according to an article at the time in The Star-Ledger of Newark. "In today's society, no one needs a semiautomatic assault weapon."

He took a similar stance two years later when he was running for the state’s General Assembly. Even as recently as 2009, when he was running for governor, Mr. Christie supported strict gun laws. “FACT: Chris Christie supports the assault weapons ban and all current gun laws,” his campaign said in a news release.

Mr. Christie has since moved away from his support of strict gun control laws and opposes banning assault weapons. His campaign website says that as governor, he has “fought to defend gun rights,” and highlights several instances in which he vetoed gun-related legislation.

In an interview on Wednesday on Fox News, he said that he had changed his mind since 1995, citing his time as the top federal prosecutor in New Jersey.

“I learned what the limitations are of these laws that people are talking about,” Mr. Christie said, “and how they much, much more greatly infringe on law-abiding citizens than they do anything to prevent crime.”

“Chris Christie personally contributed to Planned Parenthood.”

This is another claim that dates back to the early days of Mr. Christie’s political career.

In 1994, Mr. Christie was running to be a Morris County freeholder, the New Jersey version of a county commissioner.

"I support Planned Parenthood privately with my personal contribution, and that should be the goal of any such agency, to find private donations,” Mr. Christie was quoted as saying in a Star-Ledger article. The article also quoted Mr. Christie as saying “it's also no secret that I am pro-choice.”

Mr. Christie’s campaign said on Friday that Mr. Christie has no recollection of donating to Planned Parenthood.

Mr. Christie has since changed his views on abortion, and like other Republican presidential candidates, he supports cutting off federal funding for Planned Parenthood. As governor, he has vetoed funding for Planned Parenthood.

Explaining his shift on the issue of abortion rights, Mr. Christie tells a story from 1995, when his wife was pregnant with their second child and he heard the heartbeat of his unborn daughter during a prenatal visit.

“On that ride home, I said my position just is not justifiable,” he said last summer on Fox News. “That's a life, and I cannot countenance the taking of that life.”

“What he mentions all the time are the headline-grabbing things,” Mr. Christie said. “But the murder rate in Chicago is up 18 percent. The murder rate in New York is up 11 percent. And this is because this president and his attorney general do not support law enforcement.”

But those numbers are overstated.

The number of murders in Chicago increased by 12.5 percent last year, according to data provided Thursday by a spokesman for the Chicago Police Department.

The increase in New York was 4.5 percent, according to statistics released Monday by the New York Police Department.

Mr. Christie was relying on outdated statistics for New York; his campaign pointed to a Daily News article in July that said murders had risen by 11 percent in New York at that point in the year. The campaign did not say where his Chicago figure came from, but it, too, may have been old news: The Chicago Sun-Times reported in May that through the first four months of 2015, murders in Chicago had risen by 18 percent.

“It’s a problem for him, and it’s a problem obviously for the Republicans,” Mr. Trump said on MSNBC’s “Morning Joe.”

Critics of Mr. Cruz have questioned his eligibility to be president for some time. Senator John McCain, who was born in the Panama Canal Zone, faced similar questions when he ran for president in 2008.

Mr. Cruz was born in 1970 in Calgary, Alberta, to an American-born mother and a Cuban-born father. His parents were in Canada working in the oil business.

The Constitution, in Article II, requires the president to be a “natural born citizen.” But the Constitution does not define that term, which is why Mr. Cruz, and other presidential candidates before him, have faced questions about their eligibility.

The general view among legal scholars is that “anyone who acquires citizenship at birth is natural born for purposes of Article II,” as Sarah Helene Duggin, a law professor at Catholic University, wrote in 2013.

Paul D. Clement, who served as solicitor general in the administration of President George W. Bush, and Neal K. Katyal, who served as acting solicitor general in the Obama administration, took up the subject in the Harvard Law Review last year.

To understand the term “natural born citizen” as written by the framers of the Constitution, they pointed to British common law and legislative action by the First Congress.

“Both confirm that the original meaning of the phrase ‘natural born Citizen’ includes persons born abroad who are citizens from birth based on the citizenship of a parent,” they wrote.

In Mr. Cruz’s case, they concluded that “despite the happenstance of a birth across the border, there is no question that Senator Cruz has been a citizen from birth and is thus a ‘natural born Citizen’ within the meaning of the Constitution.”

Senator Marco Rubio has faced numerous attacks from his rivals for the Republican nomination over missing votes in the Senate.

But Mr. Paul, in an interview Sunday on NBC’s “Meet the Press,” went too far in criticizing Mr. Rubio and another senator who is running for president, Ted Cruz.

In 2015, Mr. Rubio had the worst attendance record in the Senate. He missed 120 of 339 roll-call votes, according to Senate voting records.

That works out to 35 percent — not “the vast majority,” as Mr. Paul said.

There have been periods in which Mr. Rubio's attendance has been notably lacking: In October and November, for instance, he missed more than 80 percent of votes. But he fared better in December, missing 28 percent.

In 2015, Mr. Cruz missed 80 votes, or 24 percent.

Mr. Paul can take credit for his own attendance record: Last year he missed only 20 votes, or 6 percent.

“Vladimir Putin is a person who has killed. He’s jailed and murdered journalists, political opponents. He bombed an apartment building as a pretext to attack the Chechens. He is responsible for the downing of that Malaysian airliner over Ukraine, because he provided the antiaircraft weaponry that was used for that.”

After President Vladimir V. Putin of Russia praised Donald J. Trump last week, Mr. Trump embraced the acclaim, calling it “a great honor to be so nicely complimented by a man so highly respected within his own country and beyond.”

But one of his rivals for the Republican presidential nomination, Senator Marco Rubio of Florida, said Mr. Trump should not be honored by the praise. In an interview broadcast Sunday on CBS’s “Face the Nation,” Mr. Rubio enumerated several treacherous acts for which he said Mr. Putin was responsible. We took a look at each of his accusations.

“He’s jailed and murdered journalists, political opponents.”

A matter of interpretation: Mr. Putin has presided over a system in which there has been systematic violence against journalists and political opponents. Rights groups say that from 2000 to 2015, about 125 journalists were killed in Russia in various contexts. Naturally, Mr. Putin has not been convicted or charged in court.

A prominent journalist, Anna Politkovskaya, was murdered in 2006 on Mr. Putin’s birthday, Oct. 7. Mr. Putin denied involvement. In February 2015, a prominent opposition leader, Boris Nemtsov, was fatally shot in Moscow. He was one of Mr. Putin’s key opponents for a decade. Suspects are in custody, but they are not believed to be the planners of the assassination. Mr. Putin said last week that he had indeed been criticized by Mr. Nemtsov, but he added: “I do not think a person must be killed for this. I will never accept this. I believe that this crime must be investigated and punished.”

“He bombed an apartment building as a pretext to attack the Chechens.”

Never proven. This theory of the apartment bombings — there were several — in the fall of 1999 was promulgated by a security service defector, Alexander V. Litvinenko, who wrote a book about it, “Blowing Up Russia.” Mr. Litvinenko and his backers were sworn opponents of Mr. Putin, throwing any dirt at him that would stick. While living in Britain, where he was granted asylum, Mr. Litvinenko was killed with a radioactive poison, and the British authorities blamed a Russian agent.

That said, terrorism has traumatized Russia for two decades. Suicide bombers have killed hundreds of Russians in dozens of attacks. Suggesting that Mr. Putin was behind the apartment bombings could also be viewed as highly insensitive to the families of people who died in those bombings and in other terrorist attacks in Russia.

“He is responsible for the downing of the Malaysian airliner over Ukraine, because he provided the antiaircraft weaponry that was used for that.”

This is the position of the United States government, although it was not fully documented in the Dutch safety board report that came out a few months ago. That account, which is the official report on the cause of a crash that killed all 283 passengers and 15 crew members on a flight to Kuala Lumpur from Amsterdam.

The report did not definitively determine that the missile that downed the plane had been launched from territory held by pro-Russian rebels. But the report was not intended to assign criminal culpability, so more information may become public in a forthcoming Dutch prosecutorial investigation.

Still, all in all, the version that Mr. Rubio puts forward is the generally accepted one of what happened, as stated by Western governments. Russia denies it.

Mr. Trump is hardly alone in thinking that the country’s infrastructure has been neglected, and there are plenty of data points he could have cited to make that point.

But the figure he cited is way off.

A spokeswoman for Mr. Trump did not respond to a question about the source of his claim.

The Federal Highway Administration publishes data on the condition of the country’s bridges. As of 2014, according to the agency, about 61,000 of the country’s 611,000 bridges were rated as "structurally deficient," which works out to 10 percent.

That rating “means a bridge is safe but in need of repair, closer monitoring or weight restrictions so that it does not become unsafe for public travel,” according to the agency.

Another 85,000 bridges, or 14 percent, were classified as "functionally obsolete," which pertains to a bridge’s design.

Combining those two ratings, 24 percent of the nation’s bridges were considered deficient. But that classification by no means indicates the bridges “are in danger,” as Mr. Trump said. And even so, that figure is far less than what Mr. Trump suggested.

Jihadists have posted on Twitter about Mr. Trump, who has called for surveillance of some mosques and a halt to admitting Muslims into the United States, according to the SITE Intelligence Group, which tracks jihadist propaganda.

The Clinton campaign pointed to comments by Rita Katz, the director of the SITE Intelligence Group, who was quoted by NBC News this month saying that Islamic terrorist groups “love him from the sense that he is supporting their rhetoric.”

But while it is possible that videos of Mr. Trump have been circulated by some Islamic State members, there is no known evidence that the group has used such videos as part of its recruitment efforts.

As Mr. Sanders tried to distance himself from Mrs. Clinton on advocating regime change, saying it sometimes leads to instability, as it did in Libya, Mrs. Clinton pointed to Mr. Sanders's history on Libya, noting his support for regime change there. Mr. Sanders was a co-sponsor of a 2011 Senate resolution, which passed by unanimous consent, that decried Col. Muammar el-Qaddafi and his regime for engaging "in gross and systematic violations of human rights" and called for "an orderly, irreversible transition to a legitimate democratic government in Libya."

Bernie Sanders has faced scrutiny for his mixed record on gun control. In this case, Mr. O'Malley gave an accurate account of some of Mr. Sanders's votes in Congress.

As the only House member from a hunting state with high rates of gun ownership, Mr. Sanders voted against the Brady Bill in 1993, which required background checks. His campaign said this fall that Mr. Sanders supported requiring background checks, but voted against the measure because it imposed a five-day waiting period to give time for the checks. (The waiting period was a temporary measure until an instant background check system was put in place.)

In 2005, Mr. Sanders voted in favor of a bill to shield gun manufacturers and dealers from lawsuits.

And when it comes to research dollars, Mr. Sanders was among those in 1996 who voted against restoring $2.6 million for research into gun-related deaths that had been cut from the Centers for Disease Control and Prevention. This month, however, he called for C.D.C. funding to research gun violence.

Mr. Sanders, of course, would prefer to highlight other votes he has taken in favor of gun control measures, including after the mass shooting in Newtown, Conn.

In October 2002, Mr. Sanders voted against the Iraq war, and one reason was a concern about what he termed then as a “unilateral American invasion” — as he said during the debate. But he also spoke about how a war would drain resources and energy from other domestic problems. In a news conference at the time, he said, “Maybe we should be protecting elderly people who have lost their life savings in the declining stock market.”

In recent weeks, Mr. Trump has described the Islamic State as the country’s “No. 1 threat,” promised to “bomb the hell out of ISIS,” and declared of terrorists: “You have to take out their families.”

At the debate, Mr. Bush argued that Mr. Trump was not a serious candidate, particularly on matters of security, saying that not long ago, Mr. Trump had expressed no interest in getting involved in the fight against the Islamic State.

Mr. Trump did not use those exact words in the interview, but he made clear that he was not eager for the United States to fight the Islamic State in Syria — at least at that point, before the terrorist attacks in Paris and in San Bernardino, Calif.

“Let Syria and ISIS fight,” Mr. Trump said at the time. “Why do we care? Let ISIS and Syria fight. And let Russia – they’re in Syria already – let them fight ISIS.”

Repeating his suggestion that Russia be left to “take care of ISIS,” he asked, “How many places can we be?”

Mr. Trump was not quite as hands-off as Mr. Bush suggested. He described the Islamic State as “really bad dudes” and affirmed, “We have to get rid of ISIS.”

And he did see a role for the United States in fighting the Islamic State, just not in Syria. “Now in Iraq, we have to do it,” he said.

The Bush campaign also pointed to another CNN interview, from July, in which Mr. Trump used the same phrasing that Mr. Bush recalled.

Asked his strategy toward the Islamic State, he said, “the situation with ISIS has to be dealt firmly and strongly,” but quickly added, “I would love not to be over there.”

The two Hispanic senators in the Republican presidential race — Ted Cruz of Texas and Marco Rubio of Florida — battled over immigration in the debate on Tuesday, competing to show who is tougher on border security. Mr. Rubio is vulnerable with conservatives on the divisive issue because of a bill he sponsored in 2013 that would have given people in the country illegally a pathway to citizenship.

After Tuesday’s debate, readers asked us to look into whether the two candidates had accurately portrayed their own records on immigration: “Cruz and Rubio exchanged conflicting claims on whether Cruz has or will ever support citizenship/amnesty for immigrants who are currently in the country illegally,” Laurence Schiffman wrote in. “Please clarify with a bit of historical perspective.”

Mr. Rubio was trying to dim Mr. Cruz’s luster with conservative voters, who have been gravitating toward Mr. Cruz in Iowa, by claiming that Mr. Cruz had also supported legalization for those immigrants. Directly challenged by Mr. Rubio, Mr. Cruz said twice, “I have never supported legalization.”

That’s not quite right. During the debate in the Senate over the bill in 2013, Mr. Cruz introduced an amendment that would have given legal status, but no possibility of citizenship, to those here illegally. At the time Mr. Cruz said such immigrants would be “out of the shadows” and eligible eventually to become permanent residents, although not citizens.

Recently Mr. Cruz, responding to Mr. Rubio, has said the amendment, which was not approved, was a “poison pill” designed to kill the entire bill.

In the debate Mr. Rubio also said, in accusing tones, that Mr. Cruz had supported a 500 percent increase in H-1B visas, which allow American employers to temporarily hire foreign high-skilled professionals, and a doubling of the number of green cards. True. Mr. Cruz did support both measures in 2013, although recently he has called for a halt to any increases in legal immigration and last week he introduced a bill to tighten restrictions on H-1B visas.

But Mr. Cruz’s charge that Mr. Rubio was trying “to muddy the waters” also seems right. Mr. Rubio has also supported big increases in green cards, and in January he sponsored a bill to as much as triple the number of H-1B visas. Mr. Rubio, confusingly, was attacking Mr. Cruz for agreeing with him.

Meanwhile, Mr. Cruz, as he pledged to ramp up deportations, presented figures on the enforcement records of past presidents that were misleading at best. He said President Obama was “releasing criminal aliens,” while President George W. Bush had deported more than 10 million immigrants and President Bill Clinton 12 million.

Mr. Cruz seems to have lumped together deportations — about 827,000 under Mr. Clinton and about 2 million under Mr. Bush, compared with at least 2.3 million so far under Mr. Obama — with a figure for migrants who were returned, mainly to Mexico, without being formally deported. The number of those “returns” has plunged under Mr. Obama because, with enhanced border enforcement, illegal immigration from Mexico has dropped to 40-year lows.

The bottom line: In this rivalry, Mr. Cruz has consistently taken a harder line against what he calls “amnesty” for people in the country illegally. Mr. Rubio has evolved. He renounced the 2013 bill, saying he concluded that immigration could not be fixed in one package. On Tuesday he said he still supported a pathway to citizenship, but one that would come after new border security and be at least 10 years long — and likely much longer.

Mr. Trump implied that family members knew of a young couple’s plans to carry out their ISIS-inspired massacre in San Bernardino, Calif., before it happened. The couple was known for target shooting, so presumably guns and ammunition could be seen around where they lived. The mother of the husband, Syed Rizwan Farook, lived with the couple and has been interviewed extensively by the authorities, but there is no indication that she knew about the plot: That is, in fact, one of the big outstanding questions. There has been no evidence to date of people's seeing pipe bombs “sitting all over the floor,” or anywhere.

Talking up his experience dealing with terrorism, Mr. Christie invoked the Sept. 11 terrorist attacks and his subsequent work as the United States attorney for New Jersey.

But he did not hold that post when the attacks occurred, or in the immediate aftermath.

On Sept. 10, 2001, President George W. Bush's administration notified Mr. Christie that he was "the president's choice and that extensive background checks on his qualifications would begin immediately,” according to an article in The Star-Ledger of Newark published Sept. 11.

Mr. Christie gave a similar account in 2003, telling The Jersey Journal, "Whatever I thought about the job when I said yes at 4:30 p.m. on Sept. 10, it was entirely different the next day.”

But Mr. Christie would not begin serving as the United States attorney for months. The White House announced on Dec. 7 that Mr. Bush intended to nominate Mr. Christie for the post.

The Senate confirmed him on Dec. 20. Mr. Christie was sworn in on Jan. 17, 2002, more than four months after the attacks.

CNN’s Dana Bash challenged Senator Ted Cruz over his vote for the USA Freedom Act. The law, enacted in June, ended the once-secret N.S.A. program that was systematically collecting activity logs of Americans’ domestic phone calls, but replaced it with a system that permits the N.S.A. to access activity logs where they are held on the phone companies’ servers.

Mr. Cruz said the old program “only covered a relatively narrow slice of phone calls … primarily land lines.” The new program “expands that so we now have cellphones” and other data.

Soon after, Senator Marco Rubio, who opposed the USA Freedom Act, jumped on the same issue. He said that the bill took “away a valuable tool” that allowed the N.S.A. to see who was communicating with whom in the hunt for hidden terrorists, and said “there is nothing that we are allowed to do under this bill that we could not do before.”

Mr. Cruz responded, “Marco knows what he’s saying isn’t true." He added, "What he knows is that the old program covered 20 percent to 30 percent of phone numbers to search for terrorists. The new program covers nearly 100 percent.”

Actually, only a small number of phone providers were participating in the old program, and cellphone records were a gap. The current public understanding is that only AT&T, Verizon and Sprint were contributing landline records in the old, secret program, and of those, only AT&T is known to have been contributing cellphone records. Under the new, open program, the government can get court orders to require any provider to make their databases available, which dramatically expands the potential dataset.

Because the purpose of the program is to analyze social links — direct and indirect — between people to find hidden terrorism cells, a more comprehensive dataset would seem to be more useful. On the other hand, the old program retained calling records for five years. The new program imposed no new data retention mandates on communications providers. (Federal Communications Commission rules require landline providers to hold onto calling records for at least 18 months, and there is no current legal retention mandate on cellphone records, though some hold onto them longer. AT&T, for example, apparently holds onto calling records indefinitely.)

But, as a larger reality check about how important any of this is in practice rather than in theory, the existing program, which dates to October 2001, has never thwarted a terrorist attack.

The reality is that Mr. Trump did not publicly come out against the war until 2004, the year after it began. “I do not believe that we made the right decision going into Iraq, but, you know, hopefully we’ll be getting out,” Mr. Trump said on “Larry King Live” in November 2004. He also supported President George W. Bush's re-election bid, after the war had begun.

Mrs. Fiorina said she wanted to bring back several generals whom she asserted had been forced out of their jobs for speaking their minds, including Mr. McChrystal and Mr. Petraeus.

But she left out the highly memorable episodes that prompted the departures of those two men.

Mr. Obama fired Mr. McChrystal in 2010 after the publication of an article in Rolling Stone in which the general and his aides spoke critically of senior administration officials. At the time, Mr. McChrystal was the top commander in Afghanistan.

Mr. Petraeus stepped down in 2012 from his position as director of the Central Intelligence Agency. The reason for his departure was also a high-profile episode: He resigned after an investigation by the F.B.I. found evidence that he had been involved in an extramarital affair.

Certainly the civil war in Syria helped the Islamic State spread its influence from Iraq into much of Syria. President Obama's reluctance to arm the rebels may or may not have contributed; however, it is far from clear it was ready to be an effective fighting force. But there were many other factors: The incompetence of the military in Iraq, which could have contained the Islamic State early on; bad intelligence about the Islamic State's abilities and intentions; and the reluctance of Sunni Arab states to join the fight. And that only begins the list.

They have been asked — and they have resisted. When Apple instituted end-to-end encryption in the operating system for the new iPhone, its chief executive, Tim Cook, said he would not create a back door or a front door for law enforcement or the N.S.A. to get access to such data. To do so would create a hole in the system that the Russians, the Chinese and others would exploit. Google and Microsoft have largely agreed. The world would be less safe, they argue, by making all of us more vulnerable to state-sponsored and private hackers.

The comprehensive immigration reform bill that Senator Marco Rubio helped write in 2013 included $40 billion to strengthen border security, including adding 20,000 Border Patrol agents. Mr. Rubio has since said that he does not support fixing immigration in one bill, but he has called for even more border security.

As Carly Fiorina spoke of her credentials in fighting terrorists, she invoked an anecdote about how, as the head of Hewlett-Packard, she provided a truck full of equipment that the National Security Agency needed. The agency's general counsel at the time, Robert Deitz, backed up her account in an interview this year with National Review. “HP made precisely the equipment we needed, and we needed in bulk,” Mr. Deitz said.

The world powers are not giving Iran money, but by easing or terminating sanctions, they would allow Iran to have access to many billions of dollars of its own money that has been frozen in overseas accounts. Much of the money came from Iranian sales of oil and other goods and has been frozen in China, India, South Korea and other countries for years.

Whether it would be as high as $150 billion is unknown. As Rick Gladstone reported in The New York Times, the Treasury Department has estimated that Iran has between $100 billion and $125 billion in foreign exchange assets worldwide, but that its usable liquid assets after sanctions relief would be much lower, more like $50 billion. Treasury Secretary Jacob J. Lew told Congress that after Iran’s financial obligations, it would have roughly $56 billion left.

The governor of Iran’s central bank said that of roughly $77 billion held abroad, only $29 billion would be usable because the rest had already been committed to petrochemical investments or as collateral for Chinese-financed development projects. But critics said those estimates are too low and are an attempt to shape the political debate.

Mr. Trump has long supported the death penalty. He made his call for the death penalty for people who kill police officers at an appearance in New Hampshire on Thursday, where he received the endorsement of the New England Police Benevolent Association.

The death penalty does exist for some federal crimes, including killing a federal law enforcement official. But it is seldom used; only three people have been executed by the federal government in the last half-century.

Those issues aside, if Mr. Trump wants the death penalty to be mandatory for people who kill police officers, that is problematic, too, according to experts. The Supreme Court ruled in 1976 that mandatory death sentences were unconstitutional.

Mr. Trump continues to assert that he is paying for his campaign. In an interview on CNN on Wednesday, he suggested that his financial independence allows him to speak his mind, unlike typical politicians who rely on campaign donors.

But Mr. Trump has become one of those politicians.

Early in his presidential bid, Mr. Trump did supply most of his campaign’s money, providing it with about $1.8 million in loans.

But in the quarter that ended Sept. 30, Mr. Trump raised about $3.7 million in individual contributions, according to a filing with the Federal Election Commission. His own contributions in that period totaled about $101,000.

In a news release, his campaign said it had received nearly 74,000 “unsolicited donations” during the quarter with an average contribution of about $50.

At a rally in Florida in October, Mr. Trump recalled how a woman sent him $7.50 along with a four-page letter.

“How do you send the seven dollars and fifty cents back?” Mr. Trump said. “You can’t. You can’t. There’s no letter you can write. It’s true. There’s no letter that you can write to that woman to say, ‘We don’t want your money.’ ”

Mr. Trump has noted that unlike his rivals, he has no wealthy-donor "super PACs" supporting him, which he says frees him from the influence of special interests. But as for his own campaign operation, as of Sept. 30, donations from people other than Mr. Trump had accounted for about two-thirds of the total funding for his presidential bid.

Calling for all Muslims to be barred from entering the United States, Mr. Trump cited a poll released in June by an organization called the Center for Security Policy. The group said that an online survey of 600 Muslims living in the United States found that 25 percent agreed with the statement that “violence against Americans here in the United States can be justified as part of the global jihad.”

Mr. Trump vouched for the group at a rally on Monday night. But the poll — conducted by the Polling Company, a Republican firm — is in no way truly representative of all Muslim Americans because of its methodology. The poll was not based on a random sample, but included only people who chose to participate, and therefore is not representative of the population being studied. In addition, some of the questions were leading and biased.

And though Mr. Trump may respect the organization that sponsored the survey, it has taken such extreme positions against Islam that it is shunned or ignored by most national security experts and policy makers.

Frank J. Gaffney Jr., the founder and president of the Center for Security Policy, has claimed that Muslims are planning to impose Islamic Shariah law in the United States; that President Obama is aligned with the Muslim Brotherhood, an extremist political movement born in Egypt; and that agents of the Muslim Brotherhood have infiltrated the United States government, the Republican Party and conservative political organizations.

The Southern Poverty Law Center describes Mr. Gaffney as “one of America’s most notorious Islamophobes” who is “gripped by paranoid fantasies about Muslims destroying the West from within.”

Similar assertions have been repeatedly made on blogs and conspiracy websites in the years since the attacks on Sept. 11, 2001 — and debunked.

Footage of people in the Israeli-occupied Palestinian territories actually celebrating the attacks is often incorrectly submitted as proof of such celebrations in the United States. It is also the root of a persistent rumor that involves Paterson, N.J., a city with a large Palestinian population.

The Herald News of West Paterson, N.J., took pains in a report published on Sept. 13, 2001 to discredit what Paterson officials were calling “hateful gossip” spread by the Howard Stern Show and other radio reports. The mayor at the time, Martin G. Barnes, visited the neighborhood in question and found quiet streets. His spokesman said: “It’s all fiction.” At the time, Paterson police officials speculated that people had been confused by footage of the celebrations in the Middle East.

Most contemporaneous accounts, such as this New York Times article from Sept. 12, 2001, note the Palestinian celebrations but make no mention of anything similar in the United States. Quite the contrary, many Muslim Americans expressed horror at the attacks and fears about retaliation.

A Washington Post article from Sept. 18, 2001, mentions rumors of “tailgate-style” celebrations on rooftops in Jersey City when the towers fell. The lead reporter on that article, Serge F. Kovaleski, who now works for The Times, said, "We did a lot of shoe-leather reporting in and around Jersey City.”

“I do not recall anyone saying there were thousands or even hundreds of people celebrating,” he said. “That was not the case, as best as I can remember."

Extensive searches of records and news articles by Times researchers have found no video evidence of such celebrations in New Jersey, as Mr. Trump asserts.

And Steven M. Fulop, the current mayor of Jersey City and a Democrat, put out a statement that accused Mr. Trump of “shamefully politicizing an emotionally charged issue.”

“No one in Jersey City cheered on Sept. 11,” Mr. Fulop said. “We were actually among the first to provide responders to help in Lower Manhattan.”

Still, the mayor was having trouble persuading some of his own constituents. On Sunday, hundreds of people commented on Mr. Fulop’s Facebook page about what they did — and didn’t — see in Jersey City that day.

“I'm from Jersey City and remember that day as if it were yesterday,” Leila Perez Sugrue wrote. “I walked right past people cheering for what happened that day.”

The Freedom Act required the National Security Agency to end bulk collection of domestic phone records by the end of this month, and created an alternative system in which the calling records would stay in the hands of phone companies but the N.S.A. would still be able to access them to analyze a terrorism suspect's social links in search of hidden associates.

Senator Chuck Schumer supported the legislation, as did most congressional Democrats. The A.C.L.U. called its passage a milestone, but said more changes were needed to curb the government’s surveillance powers.

Along those lines, some opponents of the legislation felt it did not go far enough. Senator Rand Paul, whose opposition to government surveillance is a big part of his presidential campaign, was among those who voted against the act.

Other critics had the opposite take, asserting that the bill would weaken national security. Mr. Rubio, who voted against the measure, took that position.

But that was hardly a consensus view among his fellow Republican senators.

Mr. Cruz was among 23 Republican senators to vote in favor of the Freedom Act. On Wednesday, he said he was “very proud of that legislation” because it “gives us the tools to stop bad guys while protecting our civil liberties at the same time."

At a rally in Tennessee on Monday, three days after the terrorist attacks in Paris that killed 129 people, Donald J. Trump took credit for having “predicted Osama bin Laden” and “predicted terrorism” before the Sept. 11, 2001, terrorist attacks. “I can feel it,” he explained, likening his instinct to his ability to “feel good location” in real estate.

Mr. Trump did warn, in general terms, about the threat of terrorism.

In “The America We Deserve,” a book he published in January 2000 when he was weighing an earlier presidential run, Mr. Trump wrote that he was “convinced we're in danger of the sort of terrorist attacks that will make the bombing of the Trade Center look like kids playing with firecrackers,” referring to the 1993 World Trade Center bombing that killed six people.

He devoted a chapter of the book to the subject of terrorism, focusing in large part on the threat of “biobombs." Mr. Trump warned that there was a “real possibility that somewhere, sometime, a weapon of mass destruction will be carried into a major American city and detonated.”

Contrary to his suggestion on Monday, Mr. Trump offered no specific prediction of a Bin Laden attack against the United States.

Instead, he briefly referred to the terrorist leader while discussing what he called the “haphazard, impulsive and unpredictable” nature of United States foreign policy.

Bin Laden was anything but an obscure figure at the time — Mr. Trump’s book came out a little over a year after the bombings of the American Embassies in Kenya and Tanzania in August 1998, attacks for which Bin Laden had been indicted.

“One day we’re told that a shadowy figure with no fixed address named Osama bin-Laden is public enemy number one, and U.S. jet fighters lay waste to his camp in Afghanistan,” Mr. Trump wrote in the 2000 book. “He escapes back under some rock, and a few news cycles later it’s on to a new enemy and new crisis.”

At the time that Mr. Trump’s book was published, Bin Laden was on the F.B.I.’s list of its Ten Most Wanted Fugitives.

In bemoaning the high cost of higher education, Mr. O'Malley cited 7 to 8 percent interest rates charged by the government for loans. Those numbers, according to data from the Department of Education, are higher than what most students would pay in interest these days, with the most common federal loans for undergraduate students carrying an interest rate of just over 4 percent.

But students who took out undergraduate loans several years ago are indeed paying close to 7 percent interest, as are some graduate student borrowers today.

Correction: An earlier version of this fact check rated Martin O'Malley's claim as false (red), but after some borrowers alerted us to their higher-interest loans from previous years, the claim was upgraded to in-between (yellow).

In defending her record on Wall Street, Mrs. Clinton cited a (rather obscure) piece of legislation that she introduced while in the United States Senate to clamp down on executive compensation.

While it is true that she introduced a bill called the Corporate Executive Compensation Accountability and Transparency Act in April 2008, the proposal did not go very far. The record shows that the bill was introduced, with no co-sponsors, and referred to the Committee on Finance. No further action was taken, Senate records show.

Mrs. Clinton's claim that wages have been stagnant since the turn of the century and the end of her husband's administration is accurate, according to purchasing power data from the Pew Research Center.

But in reality, inflation-adjusted wages have stalled for much longer, going back roughly four decades, the research shows.

As he tried to make the case for military reform, Mr. Sanders cited the jaw-dropping cost of maintaining nuclear weapons. Depending on the context, his numbers can be seen as accurate -- or off.

Here are the facts: The United States is projected to spend $348 billion, or $35 billion a year, to maintain its nuclear arsenal over the next decade, according to an analysis by the Congressional Budget Office of the Obama administration's nuclear plans.

The average annual salary of “welders, cutters, solderers, cutters and brazers” was $40,040 in 2014, according to data from the United States Labor Department. That is basically the same as the median starting salary of newly graduated philosophy majors, which was $39,900, according to data from PayScale Inc.

But that compares new college graduates at the start of their careers to welders at all stages of their careers. The median midcareer pay for philosophy majors was $81,200, according to PayScale. In the Labor Department data, postsecondary philosophy and religion teachers earned $71,350.

In other words, a college graduate, even in a field that is not commercially oriented like philosophy, typically earns substantially more than a welder by the time they advance beyond the entry-level point of their career.

Ted Cruz has a plan for a 10 percent flat tax on personal income. As the moderator Maria Bartiromo noted, he is also proposing a business tax. But it is not the typical kind of tax on business profits.

As Gov. John Kasich tried to note (over the objections of the moderators), Mr. Cruz’s proposed 16 percent “business tax” is a value-added tax, which businesses would pay on their profits plus their payrolls. This tax would collect $25 trillion over a decade, six times as much as the current corporate income tax and more than twice what Mr. Cruz’s income tax would generate.

It is the real core of his tax plan, and would be passed through to consumers in the form of higher prices.

Mr. Trump added that he and Mr. Putin “were stablemates, and we did very well that night.” His comments may have given the impression that he met Mr. Putin when they were both featured on an episode of “60 Minutes” in September.

But he left out an important geographic detail: Mr. Putin was interviewed in Russia, and Mr. Trump in New York. In other words, “they did not meet in a green room of ‘60 Minutes,’” as Major Garrett, the chief White House correspondent for CBS News, clarified Wednesday on “CBS This Morning.”

In his closing remarks, Donald J. Trump told voters that he is self-funding his campaign. And while it is true that he has given plenty of money to support his run, particularly early on, his most recent campaign finance filing showed that he had $3.7 million in what he termed “unsolicited contributions.” In that period, Mr. Trump himself put up about $100,000 in in-kind contributions.

Mrs. Fiorina was trying to draw a contrast with Donald J. Trump, who in the debate boasted of being “stablemates” with Mr. Putin when they were featured on the same episode of “60 Minutes.”

But in an interview on “The Tonight Show” in September, Mrs. Fiorina gave a different account of her meeting with Mr. Putin, which occurred at an economic conference in China in 2001. In fact, she described the setting of their meeting as a “sort of a green-room setting.”

While it is true that cities with Republican mayors frequently have lower inequality, it is not a perfect comparison. As Brookings and Politifact have pointed out, cities with higher inequality tend to be larger, and cities with lower inequality tend to be smaller. And there are reasons to think larger cities would have greater inequality.

For example, many larger cities have large financial services or technology sectors, which create enormous wealth, helping drive inequality. But it is not as though Republicans and Senator Rand Paul in particular oppose policies that benefit financiers and tech moguls.

Larger cities, as Politifact points out, also tend to attract wealthy people thanks to their urban amenities, but also have higher concentrations of poverty.

Defending his support for mass deportations, Mr. Trump turned to the history books. During the 1950s, the federal government apprehended hundreds of thousands of undocumented Mexican immigrants as part of an effort to turn back “the illegal Mexican labor invasion across our Southern border,” as Herbert Brownell Jr., the attorney general, said in 1955.

The push was called “Operation Wetback,” using a derogatory term for Mexicans who enter the United States illegally. (“He is called a ‘wetback’ because many of them swim or wade the Rio Grande in the Texas area to get into this country unnoticed,” Mr. Brownell explained.)

Operation Wetback has been criticized for civil and human rights abuses. But in its 1955 annual report, the Immigration and Naturalization Service declared the effort a success. “The so-called ‘wetback’ problem no longer exists,” the report said, adding, “The border has been secured.”

Though the federal government reported apprehending large numbers of undocumented immigrants, Kelly Lytle Hernández, a historian at U.C.L.A. who has studied the operation, said the number of people who were deported was much smaller than what Mr. Trump cited, calling the Eisenhower-era crackdown "a publicity stunt more than an aggressive law enforcement campaign." Still, she said that immigration raids during the time were terrifying for Mexicans living in the United States illegally, and that many were transported far into Mexico to make it more difficult and dangerous for them to return.

The largest American banks grew more rapidly in the years before the financial crisis, but they have continued to grow in the aftermath. The government has imposed a host of new rules intended to curtail risk-taking by those banks, and in theory to make sure even the largest banks can be dismantled rather than bailed out during a crisis.

These measures include requirements that banks increase the share of funding they get from capital (selling shares or retaining profits), and reduce their reliance on borrowed money.

But some experts agree with Jeb Bush that significantly higher capital requirements for the banks are necessary, and would be more effective than any other kind of regulatory intervention.

Mr. Trump gave enthusiastic support to Russia's military involvement in Syria and its confronting the Islamic State. But Russia's actions are more complicated than Mr. Trump suggested.

While Mr. Putin has said that Russia's military action in Syria is partly designed to hurt the Islamic State, early reports on the campaign showed that Russia had attacked areas in Syria where the militant group was not present, and that it was instead targeting Syrian rebels who oppose the Islamic State.

While it is true that the trade deal includes no binding mechanism for dealing with countries that engage in currency manipulation — something that could precipitate sanctions, for example — it does include commitments by signatories to avoid aggressively weakening their currencies to gain market share for their exports.

While many economists agree with Mr. Trump that currency manipulation by our trading partners costs American jobs, they often frown on the inclusion of strict anticurrency manipulation provisions in the text of trade agreements, arguing that it can be more fruitful to address the problem in bilateral negotiations with the offending country.

Finally, as Senator Rand Paul pointed out, China is not a party to the Trans-Pacific Partnership agreement. In fact, one of the administration’s arguments for passing the agreement is that it would help check China’s influence in the region, and its ability to “write the rules of the global economy.”